1885.] Functions of the Second and Third Kinds. 207 



where /(a, 6)= sin ama cos am&Aamfr + sin amfr cos amaAama, -f- 



(p(a, b)= cos ama cos am6- sin amaAama, sin am&Aamfr, tt- 



Yr(a, b) =AamaAam5 — A; 2 sin ama cos ama sin am/) cos am&,-i- 



and common denominator = 1 — W- sin 2 ama sin 2 ami. 



The theory of the two periods can be derived from the above and 

 the relation 



fsm h 



yi-A; 2 sin 2 0cZ0 = 



Jo 



JV 1- W sin 2 0rf0 + i | j ' ^Y=W^~e~^ J (1-&' 2 sin 2 0)1 / 

 - . -vl 



f sin A , V - 



or v / I^ 2 sin 2 0^0=E + ^(K , -E'), 



jo 



where i— and sin = /n — — — 



VI — k ^sm 2 



Elliptic Functions of the Third Kind. 



Ce de 



2. Again, if u be replaced by v=\ : -==■ -p and 



K F J ^ J o (l + ™sin 2 0)yi-ft 2 sin 2 



<0 be taken = amp, sin 0= sin amp, cos 0= cos amp, v'l— A^sin 2 — 

 Aamp, we have 



sinam{p+4n + 4?;P'}=sinam/; ; 

 cos am{p + 4n + 4dF' } — cos amp ; 

 Aam{p + 4n + 4iP'}=Aamp ; 



do 



where 



n= ? 



J o ( 1 + n sin 2 0)^1 — /c 2 sin 2 



J (l + »'si 



sin 2 0)v/l — /v' 2 sin 2 



k* + k'*=l, n'(l+n)=z-Jc'2, F=K'--?LlI' 



1 + ^ 



In this case the periods depend upon the relation 



d$ 



f«*» 1 d4 _f? 



J n (1 + n sin 2 0) v/l-^sin 2 ^, ~ J ( 1 + , 



. „, . .-nun' 0)v/l-/,: 2 sin 2 



