a 
TRANSACTIONS OF THE SECTIONS, 15 
On replacing 32(—1, —2, 0) circlor 3,, by 32(0,—1, 1) per 3,, (which only alters the 
value of A) and determining the constants by the conditions 4°=0, 4'=0, 47=1, 
4°=1, we find 
Avs}, { 20° +18x*?4+39r+9+ (92+27)(1,—-1) per2, 
+32(—0,-—1, 1) pers, 
+36(—1, 0,1, 0) per 4,}. 
We may also investigate the number of active quadratic, cubic, &c. terms in any 
derivation. Such a term as a*be is a quadratic term, as its derivations are found 
exactly as if a” were absent; ab’ is a quartic term, as also is a*b?; but ab’cis a cubic 
term, and so on. (Viewed in this manner extinct terms are merely linear terms.) 
The equation n"=1"—142n—2,,,,+(n—1)!+2° always holds good whenever 7* 
denotes the number of terms of any defined class in the 2th derivation of a” ; so 
that the equations of differences are always of the same form, the alterations de- 
pending on the different values assigned to the constants. The number of quadratic 
terms in the wth derivation of a is of course 1; in the zth derivation of a’ it is 
found to be 3{7—1+(1,0,—1) per3,}, and so on. The th derivation of a? con- 
tains only one cubic term; and as a verification we notice that 1+ the expression 
just written down + the number of extinct terms found previously, =P(1, 2, 3)., 
as it should be. 
On some Elliptic-transcendent Relations. By J. W. L. Guatsuer, M.A. 
. 2) 4 - 
The author remarked that every integral of the form | o P(e) cos nade, pu being 
an eyen function, gave rise to a series such as 
o(a) —o(e— 4) —$(v +4) +(e—24) + $(e+2a) — .. =A, cos my A, cos ae * 
see Quarterly Journal, vol. i, p. 316), In this way Sir W. Thomson deduced 
the theorem 
Be tm oe 0s EO hgh ea ibg, Ets) _. we? 
Fed 9x2 
5) ee sy Sy Sa 
3 TH - Srv 
=e}. 4a" cog “+e 4 epg "4+... ; 
a a a 
which, as was noticed by Cayley, is only another form of 
K i 
© (ui)= v4 (ee IRE’ H(w+K', h’). 
The number of integrals of the above form that have been evaluated is not large, 
but all that there are appear to give elliptic-transcendent formule. An interesting 
example is the integral 
tD 
COs rz T li 
ent gt dx= Pn nr 
a ete 2n 
which, as will be seen, may be regarded as a transformation of 
1 
;. cos am = Cos am (1, h')" 
The integral can be written 
is) 
