80 Colorado College Studies. 



c'T = c'l + IT, ^ c'l + OS. 

 c'l = a' cos (cr — <;') and OS — R cos {<p -{- d' ) ; 

 . \ r' = a\ cos cr cos v'' + sin c sin c'- ) + i? ( cos <p cos V — sine sin 4' ) , 

 = (jR -|- a') cos cr cos ^'' — (i? — a') sin ?■ sine'', 

 r' i?— a' . . 



cos cr COS t'' — — — ; sin cr Sin c''. 



i? + a' ' ■ R-\-a' 



But cr and <!' are the amplitudes of u and r, whence, — comparing 

 equation (29), — 



en [u -{- v) = cnu cnv — snu sn v dn [u + v). 



Now a similar formula, containing functions of u — v, may 

 of course be obtained in like manner, using an interior circle 

 that is tangent to PJ instead of to PJ'. And when the two are 

 combined in one statement, we have 



en (u ± i^) = cnu cnv T snu snv dn (u ± v). (30 



This is the so-called "Formula of Lagrange." From it is 

 derived directly 



, . > cnu cnv — cn(u ± v) 



dn (u ± V) = , , 



^ ± sn u sn V 



and thence, by substituting the value of en [u ± v), as pre- 

 viously found, 



, . , dnudnv =F k" snu snv cnu cnv ,„^ 



dn (U ± V) — :; -; ;; s ( Ol 



^ 1 — A-"' sn'u sn~v 



It has already been noticed that when any argument u is 

 in( reased by 2nK, the corresponding amplitude cr is increased 

 by nrz. Moreover, when u = 0, then also cr = 0. And since the 

 elliptic functions of u are defined by trigonometric functions 

 oi. <p, it is at once evident that since the latter have a period of 

 2-, the elliptic functions must correspondinglj- repeat them- 

 selves in periods of AK. These inferences may be tested by the 

 addition formulae just derived, and in particular it may be shown 

 that dn u has in fact the shorter period 2-Sr. 



I wish to indicate in a single paragraph, by way of appendix, 

 how easily the definitions and formulae of the present paper 



