Mr. R. Carmichael on Laplace's Equation, its Analogues, tyc. 273 



after deducting 31*8 per cent, of quartz and 2*5 per cent, of 

 red oxide of iron, gave — 



Selenium 25*5 



Mercury 74*5 



1000 



The combination Hg Se would consist of 28*38 selenium, and 

 71*62 mercury, so that the above analysis gives rather a ratio 

 which is more closely represented by the formula Hg 6 ' Se 6 . 



XLI. Laplace's Equation, its Analogues, and the Calculus of 

 Imaginaries. By Robert Carmichael, A.M., Fellow of 

 Trinity College, Dublin*. 



IT is the object of the following paper to exhibit the applica- 

 bility of imaginary symbols of operation to integration. It 

 will be evident that the employment of such symbols is in many 

 cases indispensable if we would arrive at the most general results ; 

 that where the equations to be solved possess a symmetrical 

 character, both the methods of deduction and the solutions thus 

 derived are symmetrical, and not devoid of elegance ; and finally, 

 that the process of verification is in all cases simple and obvious. 



The first four articles of the paper were published in the Cam- 

 bridge and Dublin Mathematical Journal for May 1852, and 

 are occupied with the discussion of Laplace's equation and its 

 more immediate analogues; the remainder of the paper treats of 

 the more remote analogues. In reference to the former, it may be 

 observed that it has been long recognized by mathematicians 

 that the arbitrary portions of the solutions of partial differential 

 equations can in general be expressed under three distinct forms. 

 The first gives the arbitrary portion of the general solution as 

 the sum of an infinite number of particular expressions, and 

 though open to objection, is recommended by the circumstances 

 that it is unaffected by any signs of integration, and is wholly 

 free from arbitrary functions. The second, which is due to 

 Laplace, expresses the arbitrary portion by means of Definite 

 Integrals, under the signs of which arbitrary functions occur ; 

 and while it is, in the existing state of science, generally unat- 

 tainable, yet when once arrived at, it has the advantage of enabling 

 us to determine the arbitrary functions with considerable facility. 

 The third and most common form is that which exhibits the same 

 arbitrary portion in terms of arbitrary functions without any 

 signs of integration; and, in contrast with the last, while in 

 most cases it can be obtained by the aid of the Calculus of Opera- 

 * Communicated by the Author. 



Phil. Mag. S. 4. Vol. 6. No. 39. Oct. 1853. T 



