508 
ME, J. W. L. GLAISHEE ON THE THEOET OF ELLIPTIC FUNCTIONS. 
amplitude; for on p. 113 of his ‘ Lehre von den elliptischen Integralen und den Theta- 
Functionen’ (Berlin, 1864), Schellbach finds 
( _ (1 _ «2s+l) 
So 6,0 gx= 4 cos x 2, r-2^'co.S!i+ J "' TS ' 
(43) 
We easily see that 
2KtP ( ) 
do 0 2 o gx-=."—^- cos am — ^-=4< costf+p-^p cos 3#+&c.>, . . . (44) 
and the comparison of (43) and (44) at once gives (42), since sin 2 #-J-sinh 2 a= % (cosh 2 a 
-y- cos 2#). The result (43) is also given in the ‘ Fundamenta Nova,’ p. 102. 
It thus appears that by absolutely summing, instead of transforming, in the process 
of § 11 we obtain the series of formulae which Schellbach has given on pp. 113, 114 of 
his treatise, so that all the formulae and identities which arise from the transforma- 
tions of the elliptic functions are algebraically exhibited by the method of § 11. It is 
unnecessary to write down the series of identities analogous to (42) for the other func- 
tions, as they can be easily derived as above from the values in Schellbach. It may be 
remarked that (40) is a transformation of sec am (ui,k')= cos am u, but (42) is merely a 
transformation of cos am w=cosam u. If, therefore, we perform the process of 11 in 
reverse order (i. e. starting with the trigonometrical side of the identity to be proved, 
sum the rows instead of transforming them) we obtain (42) at once. 
It appears at first sight as if Schellbach’s formula 
2/t'K 2Kx , t 
— sec am — — = sec #4-4 cos# 
7 r 7 r 1 
(-)*<?*(! +g 2 *) 
1 + 2g ,2s cos 2 x + q i * 
(45) 
gave rise to another formula for the cosine amplitude, by writing xi for x and changing 
the modulus from k to k ' ; but this, in fact, merely gives an expression already obtained ; 
for the right-hand side of (45), on writing xi for x and e for q, becomes 
which 
sech #+4 cosh x 
( — )* cosh S[x. 
cosh 2x + cosh 2 sp.’ 
= sech #+£”(,— )* 
cosh (x—sp) + cosh (x+sp) 
cosh (x—sfi) cosh (a?-f tyt) 
=sech x+X;(-Y\sech (x—sp)- 1-sech (#+5^)}. 
Formulae such as (45) are the nearest approach I have met with to those numbered 
(10) to (17) and the other expressions at the end of § 5 ; but (besides that an imaginary 
transformation is required to reduce them to these forms) they do not put in evidence 
the periodicity of the functions. 
§ 14. It is perhaps desirable to place side by side, for convenience of comparison, all 
the different forms into which one of the functions, the cosine amplitude, has now been 
thrown. Writing, as before, 
7TU 7 XU 
^ = 2K’ z= 2 K h ( l~ e K =«“'S r=e =e % 
