Mr. A. Cayley on a Transcendent Equation. 19 



that is developed in the animal body, or the carbonic acid in the 

 lungs, or the urea in the kidneys. The study of one is of as much 

 importance as that of the other. All these actions are mutually 

 dependent and answer one common purpose — the well-being of 

 the individual, 



III. On the Transcendent gd . u = - log tan (^7T + \m) . 



By A. Cayley, Esq* 



THE elliptic functions which correspond to the modulus k=l 

 reduce themselves, as is well known, to circular functions. 

 The case in question plays implicitly an important part in the 

 general theory, and it has been particularly studied by Guder- 

 mann, and by Dr. Booth in connexion with his theory of parabolic 

 logarithms. But in the absence of a notation corresponding to 

 that used for elliptic functions in general, the theory has not, it 

 appears to me, been exhibited in its proper form. The defect 

 is very easily supplied : using for am . u, to the modulus 1, the 

 notation gd . u (Gudermannian of w), then if 



u 

 we have 



— \ — ^T = logtan(^7r + |(/)), 



J COS(/> v * 



<j)=gd. u. 

 And hence, observing that the equation between u and <£ is 



, 1 + i tan d> 

 " =1 °g l--ta4 ' 

 or, what is the same thing, 



e u — 1 



tan ^=^rjn' 



and that we have 



log tan (j + g uxj = log j— 



taniwi 



tan \ ui 



, cos hui+ sin lui 



= log- f—. r-f— . 



cos £ui — sin ^ui 



e* u + e~* u — i(e* u — e~^ u ) 



— 1 ^ u + l-\-i(e u — l) 



■ 0g e w +l— i( e «— 1) 



, 1-M'tan|<f> , ._ .. 



= log - - — |^ = log e*t = 20, 



° 1 — % tan h o 



* Communicated by the Author. 

 C2 



