182 PROFESSOR STOKES, ON THE NUMERICAL CALCULATION OF A 



the greater part of the calculation consisted in passing from the values of 7r -1 G) to the corre- 

 sponding values of m. In this part of the calculation 7-figure logarithms were used in obtain- 

 ing the value of ^m, and the result was then multiplied by 3. 



A table of differences is added, for the sake of exhibiting the decrease indicated by theory 

 in the interval between the consecutive dark bands seen in artificial rainbows. This decrease 

 will be readily perceived in the tables which contain the results of Professor Miller's observa- 

 tions*. The table of the roots of the derived equation, which gives the maxima of W'\ is cal- 

 culated for the sake of meeting any observations which may be made on the supernumerary 

 bows accompanying a natural rainbow, since in that case the maximum of the red appears to be 

 what best admits of observation. 



SECOND EXAMPLE. 



19. Let us take the integral 

 2 



u 



Z I — 



= — / 2 cos (x cos0) dQ 



7T J a 



X* X* X 6 



~2» + ¥tf ~ 2 2 4 2 6 5 



(46) 



which occurs in a great many physical investigations. If we perform the operation x — twice 



dx 



in succession on the series we get the original series multiplied by — x 2 , whence 



(*7) 



d'u 1 du 



— +- — + w = 0. 

 dx' x dx 



(48) 



20. The form of this equation shews that when x is very large, and receives an increment 

 8x, which, though not necessarily a very small fraction itself, is very small compared with x, 

 u is expressed by A cos Ix + B sin §x, where under the restrictions specified A and B are 

 sensibly constant +. Assume then, according to the plan of Art. 5, 



« = e WZ1 \Ax a + Bx^+Cxy+ ...} 



On substituting in (47) we get 



-/^l \(2a + l) Ax a ~ l + {2fi + \) BxP- 1 + ...\ 

 + a 2 Ax a ~ 2 + f?BxP- 2 + ... = 0. 

 Since we want a descending series, we must put 



2a +1=0; /3 = a-l; y-jg-I ... j 

 (2(3+ l)B = \/~la i A; (2 7 + 1) C = </~^~\ p B ... 



whence 



«=-i; /3=- 



&: 



= - 4 



• Cambridge Philosophical Transactions, Vol. vn. p. 277. 



+ This integral has been tabulated by Mr. Airy from a? = 

 to x = 10, at intervals of 0.2. The table will be found in the 

 18th Volume of the Philosophical Magazine, page 1. 



t That the 1st and 3rd terms in (47) are ultimately the im- 

 portant terms, may readily be seen by trying the terms two and 

 two in the way mentioned in the introduction. Thus, if we 



suppose the first two to be the important terms, we get ulti. 

 mately U= A or U= B log x, either of which would render 

 the last term more important than the 1st or 2nd, and if we 

 suppose the 2nd and 3rd to be the important terms, we get 



ultimately u = A <T f, which would render the 1st term more 

 important than either of the others. 



