94 OF PRESSURE AND EQUILIBRIUM. 

 1-|-^ 2 1 



and if I zzs, xzz--^ — -; wheDce the hyperbolical loga- 

 rithm of any number s may be readily found. 



Scholium 1. Though the Taylorian theorem may be 

 called a universal solvent of all analytical difficulties, yet 

 considerable judgment is required, as with other universal 

 remedies, for its proper application ; and accident, perhaps, 

 rather than talent, will often point out a device which will 

 obtain from it unexpected results. There are however two 

 general observations, respecting the employment of this 

 theorem, which it will be proper to bear in mind. The first 

 is, that where several variable quantities are concerned in a 

 problem, it will be right to consider which of them is the 

 most capable of affording a converging expression for the 

 others b} a series of its powers ; thus, in the case of atmos- 

 pherical refraction, the change of density of the medium may 

 be easily obtained in terms of the refraction, supposed to be 

 given, while the series for expressing the refraction in 

 terms of the density is of little or no use ; although the 

 celebrated author of the theorem imagined, that he had 

 sufficiently solved the problem of refraction, by determin- 

 ing a few of its first coefficients. The second observation 

 is, that the employment of the theorem frequently requires 

 a beginning to be made with a series obtained by the 

 method of indeterminate coefficients ; and that it may then 

 be applied with advantage to the completion of Ihe com- 

 putation, when the series thus found loses its convergency : 

 but in this case, we must not attempt to continue the 

 series from the diflerences of its terms, since its con- 

 vergence would be little affected by this operation, but we 

 must revert to the original equation, which furnished the 

 series by a different method. Taking for an example the 



