connected with the Theory of Probabilities. 173 



before assigned, do not permit us to satisfy the system (14) by 

 positive values of x, y, s. 



Now as z varies from to cc, t will vary continuously from 

 the highest of the inferior to the lowest of the superior limits of 

 r. Of this proposition I am not prepared to offer at present a 

 perfectly general proof, though I have no doubt that I possess 

 all the elements of such a proof. I am content to state that I 

 have met with no special case in which I have not been able to 

 prove it, and by a general method ; but that a difficulty, pro- 

 bably of notation, intervenes at present between this and the 

 general proof. The following particular verification may easily 

 be applied to all the cases in which n=3. I apply it to the 

 system (15). 



If we clear of fractions the two first equations of that system, 

 and arrange the result with reference to x and y, and if at the 

 same time we equate the first member of the last equation of 

 that system to t, clear of fractions, and arrange in like manner, 

 we shall have, writing for simplicity p', q', t' for \—p, ^—q, 

 \—t respectively, 



(ap'z + dp') xy + cp'zr. — bpzy — ep = 

 (aq'z -+- dq')xy — cqzx 4- bq'zy — eq = 

 (at'z — dt)xy + ct'zx + bt'zy—et = 0. 



The elimination of x and y from this system will conduct to a 

 final equation between z and t, the solution of which will deter- 

 mine the value of t when z is given. Hence it will determine 

 the value of the first member of the final equation of the system 

 (15) when a particular value is assigned to z, while the two first 

 equations of the system are satisfied. 



Effecting, then, the required elimination, we obtain a result 

 which may be thus written, 



AD-BC=0, (18) 



the functions A, B, C, D being eliminants* having the follow- 

 ing expressions : 



- ap'z -f- dp', qJz, — bpz -\ 



aq'z + dq', —cqz, bq'z S- , 

 at'z—dt, ct'z, bt'zJ 



{cp'z, —bpz, —ep~\ 



—cqz, bq'z, —eq 



ct'z, bt'z, —et 



* I use this term with Professor De Morgan in preference to the term 

 " determinant." Mr. De Morgan's reasons, which appear to me conclu- 

 sive, will be found in a note to a very able memoir ' On some points in the 

 Theory of Differential Equations,' in a recent Number of the Cambridge 

 Philosophical Transactions. 



A = 



