with a Determination of the Limit beyond which it fails. 319 
(sy! — ys’ + at! — te +0’ — ww! +00’ — xv’) = 
(sy)? + 2sy'(— ys’ + et’ — te’ + wu! — uw’ + va’ — xv’) 
+ (ys)? — 2ys'(2t — te’ + wu! — uw’ + v2’ — a0’) 
+ (2t’)? + 22t!( —te’ + wu’ — uw’ + v2’ — av’) 
+ (t2’)? — 2te' (wu! — uw’ + va! — xv’) 
+ (wu')* + 2wu'(— uw! + ve’ — xv’) 
+ (uw') — Quv'(ve' — xv’) 
+ (v2') + 2va'(— xv’) 
+ (av!) 
(se — 2s’ + ty’ — yt’ + vw! — wo! + uz’ — vu’ = 
(s2’)? 4+ Qsz/(— 2s + ty! — yt! + vw! — we! + uz’ — 2’) 
+ (2s')? — 2es'(ty’ — yt! + vw’ — we’ + ux! — au’) 
+ (ty P+ 2ty'(— yt! + vw! — wo! + ua! — ew’) 
+ (yt)? — 2yt (vw! — we’ + ur’ — ru’) 
+ (ow? + 2vw'( — we! 4+ ur’ — ru’) 
+ (we')* — 2wv'(ux’ — ru’) 
+ (ux) + 2ux’(— ru’) 
+ (2u')’. 
Now, the double products in all these groups will be found to cancel. We 
may readily satisfy ourselves of this as follows. Commencing with the first set 
of double products, let there be written beneath the successive terms within the 
vincula the numbers 1, 2, 3, &c., in order, up to the number 28, which will fall 
beneath the term zz’, in the last vinculum: then, upon searching among the 
other groups, twenty-eight terms will be found to cancel these. Now, let the 
gaps in the second group be filled up by continuing the numbers inserted in 
the first, seeking, however, at every new insertion, for the proper neutralizing 
term among the succeeding groups; under which term, when found, the same 
number is to be written. In this way, continuing to fill up the chasms in the 
several groups, one after another, all the cancelling terms may easily be disco- 
VOL. XXI. 2u 
