Brown. — Phenomena of Variation. 527 



polation to the extent of half the observed range. It will 

 be observed that we do not say that in this example anything 

 better than is done by least squares could be done with such 

 data, but we do say that it is absolutely misleading as an 

 example of experimentation and of computation. 



23. It is now necessary to draw attention to some 

 theorems in the graphical calculus in which the combination 

 of such curves as correspond to functions of the algebraic 

 calculus is treated. X is the variable, A, B, &c, the (variable) 

 constants. Suppose we have a curve whose function is — 



F(X) =/ x (X) +/ a (X) + (other similar terms), 



then we may build up the curve of F(X) by drawing the 

 curves /(X) all to the same scale, and then adding their 

 ordinates at corresponding points of X. This is the theorem 

 of sliding, for we conceive the ordinates of each of the com- 

 ponent curves (of /(X)) to be capable of being slid over one 

 another parallel to themselves, or to the axis of Y, and we 

 so slide them that they are placed end to end, when we have 

 the ordinates of the curve of the additive function F(X) ; 

 then always, if we have found enough ordinates, we can com- 

 plete the curve by freehand drawing, or even by eye with- 

 out drawing. 



24. Next we have the theorem of one-way stretch of 

 ordinates, by which we can introduce variation in the con- 

 stants of additive functions which are linear in the said 

 constants. Thus, considering one term of the additive func- 

 tion F(X), and writing it with its constants displayed, its 

 expression is A./(X). The theorem is that if we draw the 

 curve of this function, making A take a convenient standard 

 value — say, unity — we can find the ordinates corresponding 

 to any given value of A by the use of some such device as 

 proportional compasses applied to the curve we have drawn. 

 So also with other similar terms. There is a curious point 

 with these constants which had better be pointed out to pre- 

 vent confusion — namely, that it is immaterial whether any 

 algebraic relationships (independent of X variation) exist 

 between them or not, provided that each is not fixed by 

 any combination of the others, but is capable of taking up 

 independent values. Such relationships should be studied, 

 however, with a view to facilitating the graphical work. 

 Thus, if two terms are/i(X,A) and/ 2 (X,A,B),then we may have 

 reason to prefer to take them as /^X.A) and / 2 (X,C), or as 

 / a (X,A) and/ 4 (X,B), in the latter case breaking up the second 

 function. Considerations such as these may be traced in the 

 process for Taylor's series formulae. 



25. This is all we shall need for the Taylor's series 

 analysis, but we may refer to text-books on least squares 



