242 
and the roots of the equation F,,(p)=0, be 
| Bi; Bry Bay &. 
Then it is evident that if any one of the following relations, namely, 
N+Q=), dy, az, &C., 
2N+G@=d, do, a3, KC, 
BN + A= 4), ay, a3, &C., 
&e. 
should hold ; or any one of the following, 
N+ 4, =a, a3, a, &.; N+ aQ=a, a, &e., 
2N + a= 4), az, ay, &C.; 2N+a,=a, az, &C., 
3N + % =a, a, a, &.; 3N+a,=4, a, KC., 
&e., 
—in any of these cases, some of the terms in the above solution by 
series, become infinite, and therefore the corresponding equation ceases 
to be soluble by this method, if soluble at all. 
On the other hand, if any one of the following relations, namely 
nm+a= Py, B., Bs, &e., 
2n+a=BP,, Bo, Bs, &e., 
3n+a=f,, B,, Bs, &e., 
&e., F 
should hold; or any of the following : 
n+a=f, B,, Bs &.; n+a,=f,, &e., 
2n+a,=f,, B., B3, &e.; 2n+a,=f;, &e., 
3n +a, =f, B., Bs, &e.; 8n+a,= Pf, &e., 
&e., &e. ; 
then, in the former case, the series included within the first pair of 
brackets would become, in so far as the corresponding constituents are 
concerned, terminate (if the word may be employed in its participial 
sense); in the latter case, the series included within the second pair of 
brackets would become, to a like extent, terminate. It may be noticed 
here that when an equation is said to be integrable ‘ in finite terms,’ the 
expression, as it stands, is somewhat ambiguous, but is intended to convey 
that the solution may be expressed in a determinate number of terms, as 
in contrast to an indefinite, though even converging series, or a series 
possibly reducible, but not yet reduced. 
