PREFACE. 7 



and the quantities of electricity in the bodies producing it. The 

 advantages LAPLACE had derived in the third book of the Me- 

 canique Celeste, from the use of a partial differential equation of 

 the second order, there given, were too marked to escape the 

 notice of any one engaged with the present subject, and naturally 

 served to suggest that this equation might be made subservient 

 to the object I had in view. Recollecting, after some attempts 

 to accomplish it, that previous researches on partial differential 

 equations, had shown me the necessity of attending to what 

 have, in this Essay, been denominated the singular values of 

 functions, I found, by combining this consideration with the 

 preceding, that the resulting method was capable of being ap- 

 plied with great advantage to the electrical theory, and was 

 thus, in a short time, enabled to demonstrate the general for- 

 mulae contained in the preliminary part of the Essay. The 

 remaining part ought to be regarded principally as furnishing 

 particular examples of the use of these general formulas ; their 

 number might with great ease have been increased, but those 

 which are given, it is hoped, will suffice to point out to mathe- 

 maticians, the mode of applying the preliminary results to any 

 case they may wish to investigate. The hypotheses on which 

 the received theory of magnetism is founded, are by no means 

 so certain as the facts on which the electrical theory rests; it is 

 however not the less necessary to have the means of submitting 

 them to calculation, for the only way that appears open to us in 

 the investigation of these subjects, which seem as it were desir- 

 ous to conceal themselves from our view, is to form the most 

 probable hypotheses we can, to deduce rigorously the conse- 

 quences which flow from them, and to examine whether such 

 consequences agree numerically with accurate experiments. 



The applications of analysis to the physical Sciences, have 

 the double advantage of manifesting the extraordinary powers of 

 this wonderful instrument of thought, and at the same time of 

 serving to increase them ; numberless are the instances of the 

 truth of this assertion. To select one we may remark, that 

 M. FOURIER, by his investigations relative to heat, has not only 

 discovered the general equations on which its motion depends, 

 but has likewise been led to new analytical formulae, by whose 



