464 Averages. [CHAP. xvm. 



error taken at random shall lie within that same limit, viz. t. Thus put 

 t = l-5, and we have the result -96; that is, only 4 per cent, of the errors will 

 exceed one and a half. But when we ask, one and a half what ? the 

 answer would not always be very ready. As usual, the main difficulty of 

 the beginner is not to manipulate the formulas, but to be quite clear about 

 his units. 



It will be seen at once that this case differs from the preceding in that 

 we cannot now choose our unit as we please. Where, as here, there is only 

 one variable (t), if we were allowed to select our own unit, the inch, foot, or 

 whatever it might be, we might get quite different results. Accordingly 

 some comparatively natural unit must have been chosen for us in which we 

 are bound to reckon, just as in the circular measurement of an angle as 

 distinguished from that by degrees. 



The answer is that the unit here is the modulus, and that to put = 1-5 

 is to say, suppose the error half as great again as the modulus ; the 

 modulus itself being an error of a certain assignable magnitude depending 

 upon the nature of the measurements or observations in question. We shall 



see this better if we put the integral in the form / e~ h2x2 d(hx) ; which is 



J*Jo 

 precisely equivalent, since the value of a definite integral is independent of 



the particular variable employed. Here hx is the same as x : - i.e it is 



/i 



the ratio of x to - , or x measured in terms of - . But ~ is the modulus in the 

 equation ( y = -1= e~ Wvfi ) for the law of error. In other words the nu- 



merical value of an error in this formula, is the number of times, whole or 

 fractional, which it contains the modulus. 





