384 Mr MURPHY'S THIRD MEMOIR ON THE 



Thus let the ratio of the weight to the extent of an element P of 

 a straight rod AB be expressed by the transient function 



(\-h){\+h) I, . , 



- — —^ /o -^N ■ J.2 ' when ^ = 1 ; 



and where AP=(p, and the whole length AJB = ir, and n is very great 

 and integer. 



Then the whole weight is finite, viz. f - — ^—^ — '—- — - — '—n = 1, vet 

 " J^l — 2hcos2n(p + h^ ^ 



this function has only an existence when = 0, -, — , — ...&c., and 



therefore the rod is actually composed of disjoint particles Q,, Qa, Q3, 



&c. which are separated by equal intervals, each infinitesimals, viz. -, 



when n is very great, and equal to the actual number of particles ; 

 the action of such a system on another given one, may always be 

 estimated by using the transient function in its general form, and lastly, 

 putting h equal unity. 



46. Equations which contain two or more Definite Integrals. 



Given, jj cp (t) .f(t, a, b) + f,^l.{t) .F {t, a, b) = E {a, b), 



the forms of the functions^ F, E being known, the forms of and 

 ■<\f are required. 



Put /(#, a, b) = ^oPo + A,P, + A^P^ + A^Ps + &c. ad inf. 



where A^, Ai, A2, &c. are known functions of a and b, and Po, Pi, &c. 

 any self-reciprocal functions of t, such that ftPr!^ = a„, which will be a 

 known numerical quantity. 



Similarly, F {t, a, b) = B,Po + B,Pi + B,P, + B,P^ + &c. ad inf., 

 where B^, Bi, B^, &c. are known functions of a and b. 



Again, let (p{t) =CoPo + c,Pi +CaP2 + C3P3, &c. ad inf. 

 where Co, c,, Ca, &c. are unknown numerical quantities, 

 and \l/{t) = eoPo + ejPj+e2P2 + e3P3,&iC. ad inf.. 



