531 Transactions. — Chemistry and Physics. 



the motion of the water was discontinuous ; more likely there 

 are systematic instrumental errors. (See Graph C.) 



II. 



A few remarks may be made with respect to the arrange- 

 ment of deviations in least-square form, the graphic process 

 in the case of the power-expansion formula especially 

 giving very convenient first approximations to the least- 

 square values of the constants. If we take the expression 

 " the mean" to signify that the algebraic sum of the deviations 

 concerned is zero, and " the weighted mean " the same with 

 respect to the deviations multiplied by datum values of certain 

 weighting functions, then we may define least squai - es as the 

 process which makes the weighted mean zero for all the 

 weighting functions which can be obtained by differentiating 

 the formula with regard to each of the constants separately 

 and introducing the datum values of X. By writing down the 

 equations which are needed to bring this about we obtain the 

 normal equations of least squares, and we notice a valuable 

 check on the correctness of a least-square reduction,* for in 

 the power-expansion formula we see that the mean must 

 hold, and also the weighted mean of the deviations, each 

 multiplied by the datum values of x, of x 2 , and so on to the 

 last degree ; or for x' 1 we may substitute the standard para- 

 bolic, and so on. If, considering the formula to be in 

 the standard terms, we examine a graph of deviations we 

 can easily see that to approximate to least-square form we 

 must take out all the amounts of standard components that 

 will diminish the general magnitude of the deviations, but 

 without allowing our judgment to come into play with regard 

 to the run of any systematic deviation. 



A little practice will often enable us to get such a close 

 approximation to least-square form that the solution of the 

 normal equations becomes much simplified. The normal 

 equations, again, may be found more easily solved if made 

 up in standard terms, for in examples similar to that of 

 section 18 some of the coefficients in the normal equations 

 tend to become zero, with formulae of larger degree than the 

 second — that is, using the formulae of the " Notes on the 

 Graphs." 



III. 



It is perhaps profitable to remark that, for the proper 

 appreciation of a graph, we must get rid of the confusion that 

 sometimes arises from the algebraical usage of making the 

 symbols — and -+- stand for the operations of addition and sub- 

 traction and also as signs to designate whether a magnitude 



* Given in Mr. T. W. Wright's book, page 144. 



