OF ARBITRARY CONSTANTS, &c. 119 



The integral (20) may now be cftnveniently expressed in the following form, in which the 

 discontinuity of the constants is exhibited : 



/ 47r ^.ttN^ , ,4/ 1.5 1.5.7.11 h 



\ 3 3 J I 1 . 1440?* 1.2- 144V j 



/ 27r \ ^ 1 ,* { 1-5 1.5. 7.11 1 



V 3 y 1 1.144.r* 1.2.144V j 



(25) 



to + — j denotes that the function written after it 



is to be taken whenever an angle in the indefinite series 



...0 - 47r, 0-27r, 0, + Ztt, + tv,... 

 falls within tbe specified limits, which will be either once or twice accoi'ding to the value of 0. 



17. If we put 2) = in (25), the resulting value of u will be equal to Mr Airy's integral, 



— 1 — . When = we have the 



integral belonging to the dark side of the caustic, when = ir that belonging to the bright 

 side. We easily see from (25), or by referring to Fig. 2, in what way to pass from one of 

 these integrals to the other, the integrals being supposed to be expressed by means of the 

 divergent series. If we have got the analytical expression belonging to the dark side we 

 must add + tt, - tt in succession to the amplitude of x, and take the sum of the results. If 

 we have got the analytical expression belonging to the bright side, we must alter the ampli- 

 tude of 00 by TT, and reject the superior function in the resulting expression. It is shewn in 

 Art. 9 of my paper " On the Numerical Calculation, &c." that the latter process leads to a 

 correct result, but I was unable then to give a demonstration. This desideratum is now 

 supplied. 



18. It now only remains to connect the constants A, B with C, D in the two different 

 fdrms (19) and (25) of the integral of (18). This may be done by means of the complete 

 integral of (18) expressed in the form of definite integrals. 



Let V = /" e-'^'-'^dX, 



Jo 



then ^ = - r"e-^^-'=^U3\= + cw) - ca>\ d\ 

 da? 3 Jo * * 



whence 



mx) ; 



3 3 





