io Ellery Williams Davis 



and it is thus the functions of iv that are changed into their 

 complementary functions by the transformation (.vr). 



Therefore to each geometric operation of turning over about 

 the altitude of an equilateral triangle corresponds the analytic 

 operation of taking a complement. Likewise to the geometric 

 operation of turning through 277-/3 about an axis normal to 

 the triangle and through its center, corresponds the analytic oper- 

 ation of taking complements as to two arguments in succession 

 from the trio 0, itt, iv. 



Both the group of order 8 and that of order 6 are subgroups 

 of a group of order 24. This latter group is connected with the 

 elliptic functions in a manner which we proceed to explain. 



In the theory of pendulum motion and elsewhere occurs the 

 integral. 



^ dcf> 



W- 



-s 



^o V 1—k 2 sin 2 <£ 

 Here <£ is called the amplitude of w to the modulus k. 



cf>=am (w,k,) =amw, 

 sin 4>=sin amzv=s7ize/, 

 cos <$>=cos amw=cnw, 



V 1 — k 2 sin 2 cf>=/\<fi=l\amn=d7iu. 



Glaisher has used nsw, new, ndiv to denote the reciprocals of 

 these functions, while sczv, cdw, dszv, . . . mean respectively 

 smv/cmv, enw/dnw, . . . 



It is to be noticed that if k—o, then 



w=4>, snw=sin w=sin <£, en7t'=eos Tv—eos <j>, 

 d?iw=i; scw=tan w=tan 4>, etc. 



If, however, k=i, 



gud w=4>, snzv=tan i<j>, cnw=dnzv=sec ?'<$>, etc. 

 If we have four real quantities, a, /?, y, 8, such that 

 oc>a>/?> 7 >8; 



240 



