Addition formulas of Leaf Functions and Hyperbolic Leaf Functions

0 1 √ 1−t2 dt, such as the second degree of a polynomial and the constant term 1, except for the sign − and +. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect to the leaf function through the imaginary number i, such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number. In this study, we refer to this function as the hyperbolic leaf function. By making such a definition, the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas, such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions. Using the addition formulas, we can also derive the double angle and half-angle formulas. We then verify the consistency of these formulas by constructing graphs and numerical data.

r 0 1 √ 1+t 2 dt is similar to the inverse trigonometric function arcsin(r) = r 0 1 √ 1−t 2 dt, such as the second degree of a polynomial and the constant term 1, except for the sign − and +. Such an analogy holds not only when the degree of the polynomial is 2, but also for higher degrees. As such, a function exists with respect to the leaf function through the imaginary number i, such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number. In this study, we refer to this function as the hyperbolic leaf function. By making such a definition, the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas, such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions. Using the addition formulas, we can also derive the double angle and half-angle formulas. We then verify the consistency of these formulas by constructing graphs and numerical data.
Keywords Leaf functions · Hyperbolic leaf functions · Lemniscate function · Jacobi elliptic functions · Ordinary differential equation 1 Introduction

Leaf Functions and Hyperbolic Leaf Functions
An ordinary differential equation consists of both a function raised to the 2n − 1 power and the second derivative of the function.
The preceding equation is the ODE that motivated this study. Although the equation (1) is a simple ordinary differential equation, it has a very important meaning because it generates characteristic waves. By numerically analyzing the solution that satisfies this equation, we can obtain regular and periodic waves [1] [2]. The form of these waves differs from the form of the waves based on trigonometric functions. The function that satisfies this ordinary differential equation is called a leaf function, and it describes the features of these functions. Eq. (1) is transformed as follows: * 10-3 Takiharu The preceding integral is defined as the inverse function arcsleaf n (l) of the leaf function. Another function can be defined as follows: The preceding integral is also defined as the inverse function arccleaf n (r) of the leaf function with a different integral domain compared to Eq. (2). The variable n represents a natural number, and it is referred to as the basis. Moreover, the ordinary differential equation that is satisfied by the hyperbolic functions r(l) = sinh(l) and r(l) = cosh(l) is described as follows.
Compared to Eq. (1), the difference in Eq. (4) is the positive sign on the right hand side of the equation. The inverse hyperbolic functions arsinh(r) and arcosh(r) are well known as: The contents of the root in the integrand constitute a polynomial. The polynomial of the inverse hyperbolic function and that of the inverse trigonometric function both have a degree of 2. The magnitude 1 of the constant term in the root is also the same. The difference between the inverse functions of the trigonometric function and the hyperbolic function is the sign of the polynomial in the root. Using Eqs. (5) and (6), it is seen that trigonometric functions and hyperbolic functions have relational equation through imaginary numbers. Based on this relationship, similar functions also could be paired with leaf functions though analogy relation (See Appendix D in detail). These functions are called hyperbolic leaf functions and consist of two functions. One function is defined as follows.
r(l) = sleafh n (l)(n = 1, 2, 3 · · · ) The limit exists for the function sleafh n (l) (See Appendix F). The domain of the variable l is defined as follows: − ζ n < l < ζ n The initial conditions of the preceding equation are defined as follows.
The inverse function of the hyperbolic leaf function is derived as follows: Here, the prefix a of both hyperbolic leaf functions sleafh n (l) and cleafh n (l) are defined as the inverse functions.

Comparison of Legacy functions
The leaf functions and the hyperbolic leaf functions based on the basis n = 1 are as follows: Lemniscate functions were proposed by Gauss [3]. The relation equations between these functions and leaf function are as follows: The definition of the function slh(t) in Eq.(24) can be confirmed based on references [4] [5]. A function corresponding to the hyperbolic leaf function cleafh 2 (t) is not described in the literature [6] [7]. In the case where the basis n 3, the leaf function or the hyperbolic leaf function cannot be represented by a legacy function such as the lemniscate function.

Originality and Purpose
Fagnano discovered the double formulas of the lemniscate function in the 18 th century [8]. Furthermore, based on Fagnano's formulas, Euler derived the addition formulas of the lemniscate function [9]. Historically, there has been no discussion on the basis n = 3 in Eq. (2), (3), (16), and (17). Therefore, the addition formulas of the leaf function based on the basis n = 3 were investigated[10]. However, the addition formulas of the hyperbolic leaf function based on the basis n = 3 have not been presented. In the case where the basis n = 4 or more, there is no clear description in the literature about the addition formulas of hyperbolic leaf functions. In the case of hyperbolic leaf functions based on n = 1, these functions represent the hyperbolic functions sinh(l) and cosh(l). Therefore, the addition formulas of the hyperbolic leaf function are the same as the addition formulas of the hyperbolic function. The hyperbolic leaf function based on n = 2 represents the Hyperbolic lemniscate function slh(l). There is no clear description in the literature about addition formulas of the function slh(l). In the case of a hyperbolic function with the basis n = 3, as previously discussed, there has not been any historical discussion on this subject. The purpose of this report is to propose addition formulas for the hyperbolic leaf functions with basis n = 2 and n = 3, in addition to establishing both double-angle and the half-angle formulas using addition formulas. A similar analogy exists in the relation between the leaf function and the hyperbolic leaf function such that the relation between the trigonometric function and the hyperbolic function can be derived using imaginary numbers. Using this analogy, the addition formulas of hyperbolic leaf functions based on n = 3 can be derived from the addition formulas of leaf functions based on n = 3. Using addition formulas, we present numerical data and curves derived from the hyperbolic leaf function and show that these addition formulas in the section 2 are consistent.
2 Addition formulas

Addition formulas of leaf function
Let the two variables be l 1 and l 2 . The addition formulas of the leaf functions sleaf 1 (l) and cleaf 1 (l) are as follows.
The preceding two equations have the same meaning as the addition formulas of trigonometric functions. Next, the addition formulas of the function sleaf 2 (l) can be stated as follows.
Depending on the domain of the variable l of the leaf function, the signs of both ∂sleaf 2 (l 2 )/∂l 2 and ∂sleaf 2 (l 1 )/∂l 1 change. The symbols m and k represent integers. Eq. (27) can be summarized according to the domain of variables l 1 and l 2 .

(37)
The preceding equation can be summarized as follows according to the domain of the variables l 1 and l 2 .

Addition formulas of hyperbolic leaf function
Let the two variables be l 1 and l 2 . Considering the imaginary number i, the relation between sleaf 1 (l) and sleafh 1 (l) and the relation between cleaf 1 (l) and cleafh 1 (l) can be obtained as follows(See Appendix D in detail): In Eq. (25) and Eq. (26), the variables l 1 and l 2 are replaced with the variable i · l 1 and i · l 2 , respectively.
The preceding equation has the same meaning as the addition formulas of a hyperbolic function.
Next, let us consider the case of n = 2. The relation between sleaf 2 (l) and sleafh 2 (l), and the relation between cleaf 2 (l) and cleafh 2 (l) are as follows( See Appendix D ): As shown in Eq. (49), in the case where n = 2, the function sleaf 2 (i · l) and sleafh 2 (i · l) is equal to the functions i · sleaf 2 (l) and i · sleafh 2 (l), respectively. Therefore, we cannot derive the addition formulas of sleafh 2 (l) by replacing i · l with l in Eqs. (28)˘(31). Using the relation between the function sleaf 2 (l) and the function sleafh 2 (l)(See Appendix B), the addition formulas of sleafh 2 (l) can be obtained. By substituting Eq. (99) into Eqs. (28)˘(31), the following equation is obtained.
In the Ref.
The preceding equation can be summarized as follows according to the domain of variables l 1 and l 2 .
(i) In the case where both the domains 0 l 1 η 2 and 0 l 2 η 2 , or both the domains −η 2 l 1 0 and −η 2 l 2 0 (See Appendix G for the constant η 2 .), Eq. (53) is defined as follows: In the case where both the domains 0 l 1 η 2 and −η 2 l 2 0, or both the domains −η 2 l 1 0 and 0 l 2 η 2 , Eq. (53) is defined as follows: Next, let us consider the case of n = 3. The relation between sleaf 3 (l) and sleafh 3 (l), and the relation between cleaf 3 (l) and cleafh 3 (l) are as follows (See Appendix D): In Eq. (38) and Eq. (39), the variables l 1 and l 2 are replaced with the variables i · l 1 and i · l 2 , respectively. The addition formulas of sleafh 3 (l) are defined as follows: The addition formulas of cleafh 3 (l) are defined as follows: The preceding equation can be summarized as follows according to the domain of the variables l 1 and l 2 .

.1 Double angle formulas of leaf function
In the case where the basis n = 1, the variables l 1 and l 2 in Eqs. (25) and (26) are replaced with the variable l, and the double angle can be defined as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eq. (27) are replaced with the variable l, and the double angle can be expressed as follows: The preceding equation can be summarized as follows according to the domain of variable l.
(i) In the case where the domain π2 2 (4m − 1) l π2 2 (4m + 1), Eq. (64) is defined as follows: (ii) In the case where the domain π2 2 (4m + 1) l π2 2 (4m + 3), Eq. (64) is expressed as follows: The variables l 1 and l 2 in Eq. (32) are replaced with the variable l. The double angle can be defined as follows: In the case where the basis n = 3, the variables l 1 and l 2 of Eq. (37) are replaced with the variable l, and the double angle of the function sleaf 3 (2l) can be expressed as follows: (68) is defined as follows: (ii) In the case where the domain π3 2 (4m + 1) l π3 2 (4m + 3), Eq. (68) is defined as follows: In the case where the basis n = 3, the variable l 1 and the variable l 2 of Eq. (42) are replaced with the variable l. The double angle of the function cleaf 3 (2l) is then expressed as follows:

Half-angle formulas of leaf function
In the case where the basis n = 1, the variables l 1 and l 2 in Eq. (26) are replaced with the variable l/2, and the half angle is defined as follows: In the case where the basis n = 1, the variables l 1 and l 2 in Eq. (26) are replaced with the variable l/2, and the half angle can be defined as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eqs. (28)-(31) are replaced with the variable l/2, and the half angle is defined as follows: (i) In the case where the domain π2 2 (4m − 1) l π2 2 (4m + 1) (See Appendix E for the constant π 2 ), the half angle formulas are expressed as follows: (ii) In the case where the domain π2 2 (4m + 1) l π2 2 (4m + 3), the half angle formulas are defined as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eqs. (33)-(36) are replaced with the variable l/2 and the half angle is expressed as follows: In the case where the basis n = 3, the variables l 1 and l 2 in Eqs. (38)-(39) are replaced with the variable l/2 and the half angle of the function (sleaf 3 (l) is defined as follows: (i) In the case where the domain π3 2 (4m − 1) l π3 2 (4m + 1) (See Appendix E for the constant π 3 ), the half angle is defined as follows: (ii) In the case where the domain π3 2 (4m + 1) l π3 2 (4m + 3), the half angle is expressed as follows: In the case where the basis n = 3, the variables l 1 and l 2 in Eqs. (60)-(61) are replaced with the variable l/2 and the half angle of the function cleaf 3 ( l 2 ) is defined as follows:

Double angle formulas of hyperbolic leaf function
In the case where the basis n = 1, the variables l 1 and l 2 in Eqs. (47) and (48) are replaced with the variable l and the double angle can be expressed as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eq. (52) are replaced with the variable l, and the double angle is defined as follows: The variables l 1 and l 2 in Eq. (53) are replaced with the variable l. The double angle is then defined as follows: In the case where the basis n = 3, the variables l 1 and l 2 of Eq. (58) are replaced with the variable l, and the double angle of the function sleafh 3 (2l) is defined as follows: In the case where the basis n = 3, the variables l 1 and l 2 of Eq. (59) are replaced with the variable l, and the double angle of the function cleafh 3 (2l) is defined as follows:

Half-angle formulas of leaf function of hyperbolic leaf function
In the case where the basis n = 1, the variables l 1 and l 2 in Eq. (48) are replaced with the variable l/2, and the half angle is defined as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eq. (52) are replaced with the variable l/2, and the half angle is defined as follows: (i) In the case where the domain |l| |ζ 2 | (See Appendix F for the constant ζ 2 ) (See Appendix H for the periodicity n = 2), the half angle formulas are expressed as follows: (ii) In the case where the domain |ζ 2 | |l|, the half angle formulas are defined as follows: In the case where the basis n = 2, the variables l 1 and l 2 in Eq. (53) are replaced with the variable l/2, and the half angle can be expressed as follows(See Appendix F for the constant ζ 2 ) (See Appendix H and the periodicity n = 2): (i) In the case where the domain |l| |η 2 |, the half angle formulas are defined as follows: (ii) In the case where the domain |η 2 | |l|, the half angle formulas are defined as follows: In the case where the basis n = 3, the variables l 1 and l 2 in Eq. (58) are replaced with the variable l/2, and the half angle of the function sleafh 3 (l) is defined as follows: In the case where the basis n = 3, the variables l 1 and l 2 in Eq. (59) are replaced with the variable l/2 and the half angle of the function cleafh 3 (l) is defined as follows: 4 Numerical analysis

Numerical analysis of leaf function
In Eq. (1), the graph of the leaf function at the basis n = 1 is shown in Fig. 2 and Fig. 3. Curves of the functions sleaf 1 (l) and cleaf 1 (l) are shown in Fig. 1 and Fig. 2. The horizontal and the vertical axes represent the variables l and r, respectively. Numerical data for the leaf functions sleaf 1 (l) and cleaf 1 (l) are summarized in Table 1. These data are obtained using Eq. (2) and the Eq. (3). The curves are the same as those of the trigonometric functions r = sin(l) and r = cos(l). Using the addition formulas of Eq. (25) and Eq. (26), the curves of the leaf functions sleaf 1 (l) and cleaf 1 (l) are translated in the direction of the axis l. Fig. 4 shows graphs of the double angle sleaf 1 (2l) and the half angle sleaf 1 (l/2). These data are obtained based on Eq. (62) and Eq. (72). Fig. 5 shows curves for the double angle cleaf 1 (2l) and the half angle cleaf 1 (l/2). These data are obtained based on Eq. (63) and (73), respectively. The amplitude of the wave is 1 where one period is 2π 1 (= π = 2 × 3.1415 · · · ).
Next, the graph of the leaf function of the basis n = 2 in Eq. (1) is shown. The curves of the leaf functions sleaf 2 (l) and cleaf 2 (l) are shown in Figs. 3 and 4. Numerical data for these two leaf functions are summarized in Table 1. These curves are the same curves as those of the lemniscate elliptic functions r = sl(l) and r = cl(l). Using the addition formulas of Eq. (1), the curves of the leaf functions sleaf 2 (l) and cleaf 2 (l) are translated in the direction of the axis l. Fig. 8 shows graphs of the double angle sleaf 2 (2l) and the half angle sleaf 2 (l/2) obtained using Eqs. (65) and (74)-(75). Fig. 9 shows graphs of the double angle cleaf 2 (2l) and the half angle cleaf 2 (l/2) obtained using Eqs. (67) and (76). The amplitude of the wave is 1 and one period of the function cleaf 2 (l) is 2π 2 (= 2 × 2.622 · · · ). Next, the graph of the leaf function of the basis n = 3 in Eq. (1) Fig. 12 shows graphs of the double angle sleaf 3 (2l) and the half angle sleaf 3 (l/2) obtained using Eqs. (68) and (77) -(78). Fig. 13 shows graphs of the double angle cleaf 3 (2l) and the half angle cleaf 3 (l/2) obtained using Eq. (71) and (79). The amplitude of the wave is 1 and one period of the function cleaf 3 (l) is 2π 3 (= 2 × 2.429 · · · ).

Numerical analysis of hyperbolic leaf function
The graph of the hyperbolic function obtained using Eq. (4) is shown. Curves of the leaf functions sleafh 1 (l) and cleafh 1 (l) are shown in Figs. 14 and 15. The horizontal and vertical axes represent the variables l and r, respectively. The numerical data of the leaf functions sleafh 1 (l) and cleafh 1 (l) obtained using Eq. (4) are summarized in Table 1. Figure 2: Translation of the curves of the function sleaf 1 (l) obtained using the addition formulas based on the basis n = 1 These curves are the same curves as those of the hyperbolic functions r = sinh(l) and r = cosh(l). The curves of the leaf functions sleafh 1 (l) and cleafh 1 (l) are translated in the direction of the axis l. These data are obtained using the addition formulas of Eqs. (47) and (48). Fig. 16 shows graphs of the double angle sleafh 1 (2l) and the half angle sleafh 1 (l/2) obtained using Eqs. (80) and (86). Fig. 17 shows the graph of the double angle cleafh 1 (2l) and the half angle cleafh 1 (l/2) of the leaf function cleafh 1 (l) obtained using Eqs. (81) and (87).
Next, the curves of the leaf functions sleafh 2 (l) and cleafh 2 (l) are shown in Figs. 18 and 19. The horizontal and vertical axes represent the variables l and r. The numerical data for the leaf functions sleafh 2 (l) and cleafh 2 (l) obtained using Eqs. (5) and (6) are summarized in Table 2. Using the addition formulas of Eq. (52) and the Eq. (53), the curves of the leaf functions sleafh 2 (l) and cleafh 2 (l) are translated in the direction l. Fig. 20 shows graphs of the double angle sleafh 2 (2l) and the half angle sleafh 2 (l/2) obtained using Eqs (82) and (88). Fig. 21 shows graphs of the double angle cleafh 2 (2l) and the half angle cleafh 2 (l/2) obtained using Eqs. (83) and (90). Limits exist for the functions sleafh 2 (l) and cleafh 2 (l), respectively. (See Appendix F and Appendix G).
Next, the graph of the hyperbolic function at the basis n = 3 in Eq. (4) is shown. Curves of the leaf functions sleafh 3 (l) and cleafh 3 (l) are shown in Figs. 22 and 23. The horizontal and vertical axes represent the variables l and r, respectively. The numerical data of the leaf functions sleafh 3 (l) and cleafh 3 (l) are summarized in Table 2. Using the addition formulas of Eq. (58) and the Eq. (59), the curves of the leaf functions sleafh 3 (l) and cleafh 3 (l) are translated in the direction l. Fig. 24 shows graphs of the double angle sleafh 3 (2l) and the half angle sleafh 3 (l/2) obtained using Eq. (82) and Eq. (88). Fig. 24 shows graphs of the double angle cleafh 3 (2l) and the half angle cleafh 3 (l/2) obtained using Eqs (83) and (93). Limits exist in the functions sleafh 3 (l) and cleafh 3 (l), respectively. For the function sleafh 3 (l), the limit exists at ±ζ 3 (See Appendix F for the constant ζ 3 ). The curve of the function sleafh 3 (l) monotonically increases in the domain −ζ 3 < l < ζ 3 . In the case of the function cleafh 3 (l), the limit exists at ±η 3 (See Appendix G for the constant η 3 ). The domain of the function cleafh 3 (l) is −η 3 < l < η 3 .

Conclusion
Based on the analogy between the trigonometric and hyperbolic function, the hyperbolic leaf function paired with the leaf function was defined. The main conclusions can be summarized as follows: · The relation equation between the leaf function and the hyperbolic leaf function are derived using imaginary numbers. Figure 3: Translation of the curves of the function cleaf 1 (l) obtained using the addition formulas based on the basis n = 1 · The addition formulas of the hyperbolic leaf function can be derived by using addition formulas of the leaf function with the basis n = 1, 2, 3.
· In both the leaf function and hyperbolic leaf function based on the basis n = 1, 2, 3, half angle and double angle formulas are derived using addition formulas As a future research topic, we will investigate whether the periodicity of the hyperbolic leaf function exists. In the case where the basis is n = 2, a limit exists in hyperbolic function. By appropriately setting the initial conditions, the addition formulas n = 2 can be applied in all domains over the limit. Although the periodicity of the hyperbolic leaf function n = 2 is evident, questions remain concerning the periodicity of the hyperbolic leaf function n = 3. In the case where the basis is n = 3, a limit also exists for the hyperbolic leaf function. However, the addition formulas of the hyperbolic leaf function cannot be applied outside of its domain. At basis n = 3, the periodicity of the hyperbolic leaf function is not observed. Another unaddressed issue is that the addition formulas of the leaf function based on the basis n = 4 or more are not known.   Figure 4: Curves of the function sleaf 1 (l), sleaf 1 (2l) and sleaf 1 (l/2) based on the basis n = 1 Table 3: Values of constants π n n π n 1 3.1415926535 · · · 2 2.6220575542 · · · 3 2.4286506478 · · · · · · · · · (sleaf 2 (l)) 2 + (cleaf 2 (l)) 2 + (sleaf 2 (l)) 2 · (cleaf 2 (l)) 2 = 1 The relation equation between the hyperbolic leaf function sleafh 2 (l) and the hyperbolic leaf function cleafh 2 (l) is as follows [12] [11]: The relation equation between the hyperbolic leaf function cleaf 2 (l) and the hyperbolic leaf function cleafh 2 (l) is as follows: Table 4: Limits ζ n of the hyperbolic leaf function sleafh n (l) n ζ n 1 Not applicable · · · 2 1.8540746773 · · · 3 1.4021821053 · · · · · · · · · -1.5 Figure 5: Curves of the functions cleaf 1 (l), cleaf 1 (2l) and cleaf 1 (l/2) based on the basis n = 1 Table 5: Limits η n of the hyperbolic leaf function cleafh n (l) n η n 1 Not applicable · · · 2 1.3110287771 · · · 3 0.7010910526 · · · · · · · · · cleaf 2 (l) · cleafh 2 (l) = 1 The relation equation between the hyperbolic leaf function sleaf 2 (l) and the hyperbolic leaf function sleafh 2 (l) is as follows: (sleaf 2 ( √ 2l)) 2 = 2(sleafh 2 (l)) 2 1 + (sleafh 2 (l)) 4 (99) Figure 6: Translation of the curves of the function sleaf 2 (l) obtained using the addition formulas based on the basis n = 2

Appendix D
Using the imaginary number, the relations between the leaf function and hyperbolic leaf function are described in section[12] [11]. To derive the relation between these two functions, the following equation is defined.
The symbol i represents the imaginary number. Substituting the preceding equation into Eq. (2) yields the following equation: Here, the parameter t is replaced with i · ξ (t = i · ξ). In the case where t = 0, ξ is zero. In the case where t = i · u, ξ is u. Thus, the following equation is obtaine: Let n be an odd number, that is, n = 2m − 1(m = 1, 2, 3, · · · ). The following equation is then obtained, The following equation is obtained based on the preceding equation as follows: Eq. (106) can be defined as follows: The following equation is obtained using Eq. (102) and Eq. (109).
sleaf n (l) = −i · sleafh n (i · l) Next, let us consider the case where n is an even number. In the case where n = 2m(m = 1, 2, 3 · · · ), the following equation is obtained: The following equation is the obtained: Here, the leaf function sleaf n (l) has the following relation [1]: Eq. (113) can be expressed as follows: Figure 8: Translation of the curves of the functions sleaf 2 (l), sleaf 2 (2l) and sleaf 2 (l/2) obtained using the addition formulas based on the basis n = 2 The following equation is obtained using Eq. (102) and Eq. (115): In the case where n is an even number, the following equation is also derived: Next, let us consider Eq. (3). This equation can be transformed as follows: The following equation is also obtained: The following equation is obtained using Eq. (17): The following equation is obtained using Eq. (119) and Eq. (120): cleaf n (i · l) = cleafh n (l) Alternatively, the following equation is obtained by substituting i · l into l: Figure 9: Translation of the curves of the functions cleaf 2 (l), cleaf 2 (2l) and cleaf 2 (l/2) obtained using the addition formulas based on the basis n = 2 cleaf n (l) = cleafh n (i · l) In the preceding equation, the following equation is applied: cleaf n (l) = cleaf n (−l) (123) Figure 11: Translation of the curves of the function cleaf 3 (l) obtained using the addition formulas based on the basis n = 3 The variable m represents an integer. The graph based on these definitions is shown in Fig. 25(sleafh 2 (l) ) and