1893.] On Operators in Physical Mathematics. 121 



is made larger, but the promise is not fulfilled when ra is as big as 10. 

 Take n = ^ and m = 10 in the series (41), (42), so that 



305 / /f ,i"\ 



(75) 



(76) 



Here = 1 makes M a little less than 1J, while the first term of v is 

 2'965, which is very little changed by the next two. But observe a 

 fresh peculiarity in the v series. The change from convergency to 

 divergency at the fourth term is so immensely rapid that this fact 

 alone might render the series quite unsuitable for approximate 

 numerical calculation. A portion of the term following the least 

 term might be required (though not in the last example), but when 

 this term is a large multiple of the least term, no definite information 

 is obtainable. 



What is the Meaning of Equivalence ? Sketch of Gradual Development 

 of Ideas concerning Equivalence and Divergent Series (up to 49). 



43. In the preceding, I have purposely avoided giving any de- 

 finition of " equivalence." Believing in example rather than precept, 

 I have preferred to let the formulae, and the method of obtaining 

 them, speak for themselves. Besides that, I could not give a satis- 

 factory definition which I could feel sure would not require subse- 

 quent revision. Mathematics is an experimental science, and 

 definitions do not come first, but later on. They make themselves, 

 when the nature of the subject has developed itself. It would be 

 absurd to lay down the law beforehand. Perhaps, therefore, the best 

 thing I can do is to describe briefly several successive stages of 

 knowledge relating to equivalent and divergent series, being approxi- 

 mately representative of personal experience. 



(a). Complete ignorance. 



(6). A convergent series has a limit, and therefore a definite 

 value. A divergent series, on the contrary, is of infinite value, of 

 course. So all solutions of physical problems must be in finite terms 

 or in convergent series. Otherwise nonsense is made. 



The Use of Alternating Divergent Series. Boole's Rejection of 

 Continuous Divergent Series. 



44. (c). Eye-opening. But in some physical problems divergent 

 series are actually used for calculation. A notable example is 

 Stokes's divergent formula for the oscillating function J n (a?). He 

 showed that the error was less than the last term included. Now 



