350 
mi. E. W. BAEXES OX THE THEORY OF THE 
theory are obtained solely by this course. Inasmuch as the function ro(?/i 2 : aj|, oi^) 
/n 
(0.2 
m 
tlie multiplication theory can 
can be ex|)ressed in terms of the function 
[ 
he deduced from the theory of transformafion. 
expression is in every case almost equal to that of obtaining the results ah initio, we 
adopt the latter course. 
As the work of ohtainino’ the new 
Mid tip lie oAion Theo rij. 
§ 62. We have from the detinition (§ 19) 
= - 2 
00 X 
o s' V 
and thereli»re 
hid (a;.,:) = — 2 X X 
u,=o + + WoWnf 
00 X 
)iii = (()j(2 = 0 / .. _j_ wqcoj ^ finfOn 
rn 
lit 1 til \ QQ 
—_OX X X X 
■III 
1 
V 
'■ = 0 ® = 0 ’O = 0 = 0 I + Vhfo. + m.oo., i 
in in ~ ~j 
hi — I Til — 1 
— X X Ci) I . 
= 0 S = 0 \ ''hi 
the parameters Ijeing understood to l)e coy and co.^ vlien not explicitly written. 
Integrate with respect to s and we obtain 
m~xfj.d^'> {mz) = X X ) + ^’- 
,, = 07 = 0 ' \ hi , 
where r is constant willi respect to c. 
N ow 
(-) = — y-n ("i, on) + , + X X' 
1 
1 
(r+0)3 03 
(i-)> 
Substitute from this relation in tlie identity (1), take the same number of terms 
involving 2 on both sides of tlie ec[uality, and remember that -— ■ — is 
(- + ny- 
ahvays to be regarded as a single entity. We tind that, in the limit Avhen ii is 
intinite, we have 
- r + nr X X' - == nr ^ 
0 0 H" 
rdii + i» — \ —1 1 
X' A 
d; A -' 
wliere D. = iiiyojy + iii. 
Now we have seen that (§ 22), in the limit wlien n is inlinite. 
