OF 
fx 
fa 
UNDER THE FORM OF AN IMPROPER CONTINUED FRACTION. 
thus ex.gr. the cumulant may be represented by the determinant 
475 
5-1 1 0 0 
1 5-2 1 0 
0 1 ^3 1 
0 0 1 ^4, 
and ^4 q^ q^ will in like manner be represented by the determinant 
^4100 
1 ^3 1 0 
0 1 5-2 1 
0 0 1 
which is equal to the former. 
Art. (j.) Let it be proposed in general to find the first differential coefficient in respect 
to X of the fraction 
hi 
[?i?2 %•••?«] 
where each 5- is a function of one or more variables. 
I find that the variation of F, may be expressed as follows : — 
— ••• qi-2, 
+ ^2, 93-.-9t-25 qn-^-iqn, ?«-.]'+ &C. ^ 2 , ?3---^i-2, ?t- 1] • [?«, ?«-2 • • • } 
^2, 5'3...^„]^ 
Art. {k.) Suppose i=2, and q,=a,x-\-b, q^=a^x-\-b^ q^=za^x-\-h„, 
we shall have by virtue of the above equation, 
^ p, . J J. l_ y\ 
dx 25 *• dx'\qi- q^— qs''qn] 
^ 2‘ [^5'm? ll “1“ &C. 5're— 15 qn—2 ••• q^ }• 
If we call ^2=^ every such quantity as qn-i“-qi\ represents to a constant 
factor pres the {i — l)th simplified residue {<px counting as the first of them) to 
^x 
and making certain obvious but somewhat tedious reductions, and rejecting the 
common factor obtain the expression 
Co-R? 
+ + p ^>r = {(px.fx-(p’xfx), 
Cl ' C1.C2 
C,.C3 
where Rj R2...R„ represent <px and the successive simplified residues to fx, <px, and 
