400 Mr airy, on the INTENSITY OF LIGHT 



wish to retain, even though the divergence of the series be just 

 beginning, tlie use of these terms will give the integral required with 

 the utmost practical accuracy. 



Now when vi = + SO nearly, it is found that the divergence com- 

 mences at v., or before it; and that the term v- is not so small that a 

 (Quantity which is likely to be a sensible portion of it can be safely 

 neglected. To approximate here, I have used the following considera- 

 tion. It is known that the slowly converging series 



A- B +C - D + E - F + hc. 



may be converted into a series of the same kind with much smaller 

 terms by putting it in this form, 



+ \{A-B)-\{B-C) + \{C~D)-^{D-E) + \{E-F)-kc. 



In the instance before us, such a series would be produced by commencing 

 the integration by parts with ^ {v^ - v.^ instead of v^ ; and the residual term 

 will be of the form of 



i L cos u . ^ {v,. - v„+4 or 1 /„, sin n ^ (v„ ~ v,.^,). 

 Now if the progression is stopped at such a point that -^ is, for all 



the following values of u\ greater than ,"'^\ then the quantity -r- (i^„-r„+,) 

 ^ *= dw ^ ■' aw '' 



has always the same sign, and the reasoning above shews that the residual 



integral will be less than ^ {v„ - i^.+s). A close approximation therefore 



will be obtained by summing the series as if we had begun with ^(vo — v-,) 



instead of *>„ : and this, it is easily seen, will be effected by taking half of 



the last term in each of the series multiplying sin u and cos ^^ The 



multipliers which I have used are, in fact, 



-Vo + i\ — Vi + i*'o for sin u, 



— Vi +i\ — Vi + ^V; for cos u. 



The doubt which remains extends, I apprehend, to digits in the fifth 

 place of decimals, but no higher. 



