24] ON THE INVERSE ELLIPTIC FUNCTIONS. 155 
where the extreme values of n are infinite as compared with p. This may be reduced to 
- sin^ {a; - (p + £) «} -r sin (p + £) <», 
|f[(p+W«-*] : [(*+«»] -- 
= e v -r- e u = € » t 
neglecting the exponentials whose indices are infinitely great and negative. Observing 
the value of /3 this becomes e ~ ul3x , and we have 
j 
7-/3 (•£ + = e ^ x ■ A 'g~fi • x : 
a result of the form of that which would be deduced from the equations <y_p x = e^y^ x, 
g_fix = e^gpx, <y^{x+ = Ag^x. It is scarcely necessary to remark that y_^x has the 
same relations to the change of x into x + ^ as y^ x has to that of x into x + ~. 
& <L 
20 
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