58 REMARKS ON AN ALLEGED PROOF OF 



bability of an error r may be expressed by the function ,/(r 2 ) 

 or /(#* + ;*/*), the origin of co-ordinates being placed at the mark. 

 It is of course supposed that equal errors in all directions are 

 equally probable. We have now only to determine the form 

 of/. This the reviewer accomplishes in virtue of a new as- 

 sumption, namely, that the observed deviation is equivalent to 

 two deviations parallel respectively to the co-ordinate axes, 

 " and is therefore a compound event of which they are the sim- 

 ple constituents, therefore its probability will be the product 

 of their separate probabilities. Thus the form of our unknown 

 function comes to be determined from this condition, viz. that 

 the product of such functions of two independent elements is 

 equal to the same function of their sum." Or in other words, 

 we have to solve the functional equation 



But it is not true that the probability of a compound event 

 is the product of those of its constituents, unless the simple 

 events into which we resolve it are independent of each other ; 

 and there is no shadow of reason for supposing that the oc- 

 currence of a deviation in one direction is independent of that 

 of a deviation in another, whether the two directions are at 

 right angles or not. Some notion of an analogy with the 

 composition of forces probably prevented the reviewer from 

 perceiving that, unless it can be shown that a deviation y 

 occurs with the same comparative frequency when x has one 

 value as when it has another, we are not entitled to say that 

 the probability of the concurrence of two deviations x and y 

 is the product of the probabilities of each. "Without this sub- 

 sidiary proof, the rest of the demonstration comes to nothing. 

 The conclusion to which it leads is in itself a reductio ad ab- 

 surdum. Of the above written functional equation the solution 

 is f(x*} = e mx \ m being a constant, so that the probability of 

 an error of the precise magnitude a? is a finite quantity; and 

 I need not point out to you that it follows from hence, that 

 the probability of an error whose magnitude lies between any 

 assigned limits is equal to infinity, a result of which the 

 interpretation must be left to the reviewer. He may have 

 thought that the exponential factor is the essential part of the 

 expression 



