ERRORS OF OBSERVATION. 425 



But ma may be written as ndm; and the probability of the point defining a distance 



a 



between x and ai + dai is (p{w + 6dx)nda) divided by / (p{a) + 6dx)ndx', whence, by principles 







a 



common to all questions of integration, we deduce (pxdx divided by / (px dx. Let the pro- 







blem be proposed as I have stated it, and I will defy any one to produce a solution without 

 either the distinct recognition of indivisibles which I have made, or an assumption which 

 hides it, something which " of course we may suppose." 



In elementary writing the difficulty can often be avoided, and not merely evaded; 

 especially the difficulty of the introduction of the differentials, and of the management of 

 the differential of that quantity which is a function of the others. If \^y and (px be the 



moduli oi y and x, the modulus of a? + y is j -^(q - x) (px dx, which, multiplied by dq, 



represents the probability that x +y shall lie between q and q + dq. This may be obtained 

 as follows. That the variables shall lie between x + dx and y + dy has the probability 

 (bwy^y dx dy, and notions of integration with which we are perfectly familiar, and chiefly by 



geometrical application, give flcpxdx f y^ dy\ ior the probability that a?+y shall lie 

 between p and q. If f^ydy = \//,y, this is 



/ U'M -oo)- y\,^{p - x)\<pxdx. 



The probability that x + y shall lie between q and q +dq is the differential of this with 



respect to q or j y\f{q — x)(pxdx x dq. In the same manner we obtain the following form, 



which is more convenient in some respects than that commonly given. If 0a be the 

 modulus of x„ the probability that x^ +...+ x, shall lie between q and q + dqis 



M j j (ps(q-'Bs-\)(j>s-\(.^s-i - ^s-s) (p.i(x.i - Xi) ^^Xidx,.T^ dx^dxi. 



If those notions, sound or unsound, clear or confused, on which a point has been connected 



with a line, and a line with an area, as its indivisible, were carried into the consideration of 



dv 

 magnitude in general, then — would be called the indivisible of y. This would more than 



halve the number of letters in differential coefficient; but, independently of substantial 

 objections to the introduction of the notion into elementary writing, a greater abbreviation is 

 wanted. The length of the most common words is a serious obstacle, especially in teaching : 

 and no body of educated men ever had the sense of the people at large, quem penes arbiirium 

 merely because they choose that it shall be so. Popular usage will in course of time cut 

 down the excellent words — excellent, because they say exactly what they mean — numerator 



54 — 2 



