INVERSE METHOD OF DEFINITE INTEGRALS. 383 



which contains an infinite number of constants multiplied by functions 

 of a, which may vanish or not, and be connected or unconnected ac- 

 cording both to the nature of the particular operation and the nature 

 of the calculus in which it is employed ; this has been already shewn 

 by Mr Peacock*, and in Art. 20. Sect. vi. of this Memoir. The same 

 remark applies to the value of (t) in the general equation 



to complete it we must add ^{t) where ft^{t) .f{t, a) = 0. 



To obtain \|/(/) in the equation ft<p{t) .cos (at) = F {a) above mentioned. 



Let us suppose (pi {t), (p-^ {t), found by the method of Art. 44., to 

 satisfy the equations 



Jt(px (t) . cos (at) = 1 for continuity, 

 ft(p2{t) . sin {at) = 1 for discontinuity, 



differentiating with respect to a, the first 2n times, the second 2«— 1 

 times, we get 



f,<pi{t).f"' cos (at) = 0, 

 ft (p.2 it), t"-' cos {at) ^ 0. 



Hence, 



^{t) = (p,{t) {At + Bf + Ct\ &c.} + 0,(0 {A'f + Bt^ + C't'^c.}, 

 where A, B, C, &c. A', B', C, &;c. are absolute constants. 



When transient functions appear in the appendage or even in the 

 prime solution, they must not be neglected (particularly in the mole- 

 cular investigations) except they are inadmissible by the nature of the 

 particular question, for they have a physical as well as a geometrical 

 meaning, as they are capable of expressing in continuous analytical 

 forms, the state of bodies and their mutual actions when they are com- 

 posed of absolute mathematical centres of forces, all separated mutually by 

 infinitesimal intervals. 



Q/ Q; a> 04 Qf Jr 



* Third Vol. Report of British Assoc, p. 212, &c. 



