760 
MR. J. W. L. GLAISHER ON RICCATI’S 
series terminate in consequence of the occurrence of zero factors in the coefficients of 
the terms, but if they are continued, zero factors occur also in the denominators, so 
that, after a finite number of zero terms, the series may be regarded as recommencing 
and extending to infinity. If the terminating series are supposed to recommence in 
this manner, so that all the series extend to infinity, then the relations between the 
particular integrals are the same as when p is not an integer; but if the series are 
supposed to terminate absolutely when the zero terms occur, the relations are quite 
different. As the finite portions of the particular integrals satisfy the differential 
equation, it is more natural to regard the series as terminating absolutely, and on this 
supposition the relations between the particular integrals exhibit a remarkable diversity 
of form according as p is or is not an integer. 
The second section contains what is believed to be a new form of the solution of (l) 
in the case of p = an integer. It is shown that if p = i, a positive integer, this 
equation is satisfied by the coefficient of h l+l in the expansion of in ascending 
powers of h. The six particular integrals given in § I. of the equation (1) and the 
relations connecting them are obtained by different expansions of this expression. 
The third section contains the six particular integrals of (3) and (4) corresponding 
to those of (1), from which they are deduced by means of the transformations stated 
above. 
The fourth section relates to the particular cases in which the differential equations 
admit of integration in a finite form. If a differential equation is satisfied by an 
infinite series, and if for certain values of a quantity involved in it the series termi¬ 
nates, then in this case we may present the integral in a different form by commencing 
the finite series at the other end, and writing the terms in the reverse order. 
Thus, for example, a particular integral of (1) is u = P, where 
1 -PxPppp T7+*o. 
V P(P~\) 2! P(P~i)(p- 1) 3! 
but, if p = a positive integer, then commencing the series at the other end, 
2W 
P=(-)> 
0 + 1 ) 0 + 2 ) . 
_ p(p + 1) 1 (p- l)j?Q + l)0 + 2) 1 
2 p [ 2 ax' 2.4 a~x 2 
+(-y 
1.2 . , 
. . 2 p 
1 1 
2.4 . . 
• 2 p 
aPaff 
These reverse forms in the case of the equations (1), (3), (4) are given in this 
section. 
It is worthy of remark that if we are given a particular integral of a differential 
equation in the form of a terminating series, such as, for example, 
