892 Mr MURPHY'S THIRD MEMOIR ON THE 



PAGE 



Art. 15. The reciprocal functions expressed by the general formulae for Q,, q, all possess 

 a common property, viz., their integrals vanish when taken between limits which 

 render the functions maxima or minima 333 



Art. 16. To find the complete integral of the equation 



i<'^ + (m+l)(l-2/) -^^+n(n+2m + l)u = 335 



Art. 1 7. To find explicitly the omitted part of the complete integral in Laplace's equation, 

 for the coefficients in the expansion of the reciprocal of the distance between two 

 points in a plane 338 



Art. 18. When m = —\ the general equation of Art. l6. represents the trigonometrical 

 functions 342 



Art. 19. Remarkable properties of the functions 



G = g^cose cos(a;sin0), 0'= £■>; ™s e sin (a; sin 0) 342 



Art. 20. Application to the general differentiation of rational and integer functions oi x . . . 344 

 Art. 21. The sum of all the divisors of a number n, including itself and unity 



= fy^- 1- {sin^ sin 2^ sinn^} . cos 2n(^ 346 



TT 



Section VII. On Transient Functions. 



Art. 22. Nature of transient functions 347 



Art. 23. To find a function reciprocal to f(t, n) any given function of the variable t 



and integer n ' 348 



Art. 24. To find a function V„ reciprocal to /" 349 



Art. 25. The function V„ is transient 352 



Art. 26. To express the transient function F„ in a finite form 354 



Art. 27. Discussion of the transient function Fo ; it represents the state of a body which 



an electric spark is about to enter 356 



Art. 28. To find the quantity to which F„ is the generating function 360 



Art. 29. To expand a given function in terms of the functions F„ 36l 



Art. 30. To find a function t/„ reciprocal to (h. 1. t)' 362 



Art. 31. In a finite form U„ = H — 2_ when A = l 363 



1 .2...ndh" 



Art. 32. Properties of C7„ &&f,Uj' = x', &c S6S 



Art. 33. Discussion of the function Uq 364 



Art. 34. To expand a given function in terras of the functions I/„ 365 



Art 35. To find a function reciprocal to /" when the limits of t are and =0 366 



Art. 36. To find a function F„ reciprocal to cos"^ between the limits ^ = and <p=-ir. 366 



Art. 37. The function F„ is transient - 367 



Art. 38. To express F, in a finite form 368 



Art. 39. Means of summing a series expressed in transient functions 369 



