280 Mr. R. Carmichael on Laplace's Equation, 



To complete the discussion of equation (III.), it now only 

 remains to find its solution in the third or simple functional 

 form, which is 



W = <f> (x -f iw, y +jw, z -f kiv) +yjr(%— iw, y —jw, z — Jew), . (c 3 ) 



and whose correspondence with the third forms of the solutions 

 of equations (I.) and (II.), as well as whose coincidence with its 

 own first form, are both obvious. 



4. It is evident, that, by the adoption of an extended system of 

 operations, regulated by laws similar to those already employed, 

 the same methods of solution may be applied to the general 

 equation containing n independent variables. However, when 

 the number of operations indicated by the letters i, j, k, &c. ex- 

 ceeds three, we can no longer attach to them distinct geometrical 

 conceptions. Thus the signification of i s is purely analytical, 

 and on that account it seems unlikely that the examination of 

 the general equation would lead to any practical result. 



To adopt the language of Sir William Hamilton, the quanti- 

 ties i, i P , t E , i a , &c. are, severally, imaginary units ; that is, their 

 moduli are positive unity and their squares negative unity. If 

 for each of these quantities we had substituted the root of nega- 

 tive unity, the solutions in this form could have been obtained 

 at once. Since, however, as has been before remarked, it is pro- 

 bable that these several units bear some relation to the characters 



from the equation 



2W = (e^o — e- WJ >)(f)(xyz) + (c^d -j_ e - w»)^{xyz), 



a form can be obtained, which seems to the writer to possess somewhat 

 greater generality, viz. 4ttW= 



jo Jo 



wQ>{x+iw cosw, y+jw sinttcosu, z+kw sinw . swv)smududv 

 + 

 •j- 1 1 uflr(x+iwcosu, y+^'w sin m cost?, z+Jcwsmu.mnv)sinududv t 



and by a similar process applied to the equation of the last article, its 

 solution would be 



I I z*(a? -f iz cos «, y +jz sin u cos v) sin u du dv 

 Jo Jo 



47rV=. 



d *C 27r f 

 — - I I zty(x+izcosu, y -\- j z sin u cos v) sin ududv, 



dz Jo Jo 



From the distinct geometrical characters which we are able to assign to 

 the several symbols i, j, k, it would seem that their occurrence, so far from 

 being matter of objection, will yet be found to possess some important 

 bearing upon the problems, in whose solutions such symbols are exhibited. 



