222 Mr. R. Meldrum Stewart on the 



quantities. From the N exact condition equations (3) 

 we now proceed to determine the values of any N of the 

 unknown quantities in terms of! the ~Nn-+ q — N remaining 

 ones ; and substituting these values in the equations (4) 

 there result Nn observation equations for the determination 

 of the most probable values of Nn + g — N unknown 

 quantities. Having formed the normal equations in the 

 usual way and derived the solutions, we then find the most 

 probable values of the remaining N unknowns from the 

 exact equations (3). 



It' the corrections so obtained to the assumed approximate 

 values A, B, C . . . X 1? Y l5 Z : . . . etc. are sufficiently small, 

 this may be accepted as the definitive solution; if not, 

 we assume as new approximate values the quantities 

 A-f Aa, B + A& . . . Xx + A&'i, Y 1 -\-Ay l . . . etc. and proceed 

 exactly as before to a further approximation. 



The solution of this general problem is thus perfectly 

 straightforward and unambiguous, but is entirely different 

 from the erroneous solution given by Dr. Campbell. 



It may of course be claimed that when either q or n is 

 large (and indeed even when they are small) the rigorous 

 solution of the general case as described above is cumbersome 

 and tedious, and that in many cases more approximate results 

 are quite sufficiently accurate for all practical purposes. This 

 may readily be granted ; and Dr. Campbell's paper may 

 serve a very useful purpose in drawing attention to a fact 

 which is undoubtedly true, that when the number of obser- 

 vations is large compared with that of the unknown quantities 

 required it is comparatively easy, by the application of 

 common sense, to devise approximate methods which will 

 lead to very nearly the same results as the more rigorous 

 method of least squares, and with much less labour. It is 

 necessary, however, to exercise considerable caution in the 

 choice and application of such methods ; and the fact remains 

 that (provided we accept the principle of the arithmetic 

 mean) it is more than an even chance that the results obtained 

 by least squares are nearer the truth than any other results 

 from the same data, however obtained. And it may be worth 

 emphasizing that the statement (see pp. 191, 192, 193) that 

 there are certain classes of problems in which the method of 

 least squares is not applicable, or gives ambiguous results, is 

 quite untrue. It will also do no harm to insist once again on 

 the fact that the method of least squares makes no claim to 

 deduce the true values of the quantities sought, but only their 

 most probable values. 



Though all cases are covered by the solution of the general 



