268 Mb. EARNSHAW, ON THE VALUES OF THE SINE AND COSINE, ETC. 



It appears from this example that, as has been before remarked, a os, which for distinction I call 

 a symbolical eo, is not to be confounded with co , a mere arithmetical infinity : for the former 

 ceases not to be a function of a. 



., , r" Sin oa; , . , i. ,i i /■" Sin*- 



In (Art. 26.) then when it is said that / dx is not symbolically equal to / dx, 



^ ^ 



the assertion is grounded upon this distinction between a os and co ; and it is manifest that in 



this case, supported as it is by the example quoted from Fourier, / dw is symboli- 



cally a function of a, while / dm is not a function of a. This distinction between a ec 



and 00 is of great importance in all definite integrals where the results are understood to be 

 symbolically exact ; as they are always supposed to be when they are made use of in obtaining from 

 them other definite integrals by differentiation or integration with regard to parameters. It will be 

 very obvious to any one who examines the definite integrals which have been published, that many 

 of them have been obtained without sufficiently observing this caution with respect to symbolical 

 exactness. 



S. EARNSHAW. 

 Cambridge, 



November 9, 1844. 



