IQ EESEAKCHES ON H. 



after the substitution of 



G) = 90° — 2 i]. 



and reduction of functions of double arcs to simple ones, will be transformed into 



_ — 2(Z^ 



^^^ ^^ lP+a^ — 2a7n + 4. {am — {a' — W) dn!' ^ + 4 (a^ — V) sin} ^)] |" 

 The denominator resolved as a quadratic equation gives 



The quantity f + a^ — 2 am = ?^ + a^ — 2a? cos. I being the square of the 

 distance of the pole of the needle from the end of the transverse axis of the ellipse, 

 is always positive as well as a^ — &^ therefore the sign of the root will depend 

 principally on the term am — (a- — V), and the cases are to be considered, for 

 which different integrals are to be found, according as we have the value of \am — 



C^2 j2N-|2 ^ Q^ ^ oj, __ {^ ■\- (j^ — 2 am) {o? — }?), and the roots of the equation 



(3) are real or imaginary, positive or negative. But it is evident that both roots 

 cannot be real and positive at the same time when am > {cc — J?). The second 

 member of (3) will always be a fraction, since the numerical value of the factor 

 within the brackets cannot exceed 2 and be real, and the other factor which may 



be written - (l — — r ) i^ ^'^'^ ^ fraction less than -. Hence the roots 



will be 1st both negative, 2d both positive, and always < 1, 3d both imaginary. 

 FormulEe may be easily found for these three cases. 



1. Representing therefore by ]? and ^ the numerical value of the roots when 

 both are negative, the integral may be written 



— 2 d'^ 

 (^) ^= (2 e ay [(sm.' 4 + f) {sin.'' ^ + g')]|' 

 for 4 {a~ — If) = 2^ e^ a^, where we shall always suppose jy- > g^. 



In order to reduce the last expression to the elliptic functions, let us make 



tan. 4' = — tan. d, 



whence 



/ex 0- 2 1 (fsm.^d) 2 1 {(f + ^) cos.^ ^ -J , qV{l+q^)d^ 

 (5) Sin? 4- = -$ T-i ^^^- T = ^, , — — -i a'\p= i-^— 3' ^ . 



By means of these the integral becomes 



— 2 (g ^ + cos.'' ^)' d^ 

 {2eafq^f{<t + l)'^' A^' 

 in which 



(6) T-. 



A = >/ 1 — c^ sin.^ ^ and c^ = 



p~ 



f (1 + 2^) 

 2. When both roots are positive, and 1> p^> <f, making 



jSm.~ ^ = -^ — -„ — ?^, 



1 ff COS.'' ^ 



