394 THE PRINCIPLES OF SCIENCE. [CHAP. 



(i.), multiplying the coefficients of each equation by the 

 first coefficient, and adding together all the similar co- 

 efficients thus resulting for the coefficients of a new 

 equation ; and (2.), by repeating this process, and multi- 

 plying the coefficients of each equation by the coefficient 

 of the second term. Meaning by (sum of a 2 ) the sum of 

 all quantities of the same kind, and having the same place 

 in the equations as a 2 , we may briefly describe the two 

 resulting mean equations as follows : 



(sum of a 2 ) . x + (sum of ab) . y = (sum of ac), 

 (sum of a&) . x + (sum of b z ) . y = (sum of be). 

 When there are three or more unknown quantities 

 the process is exactly the same in nature, and we get 

 additional mean equations by multiplying by the third, 

 fourth, &c., coefficients. As the numbers are in any case 

 approximate, it is usually unnecessary to make the com- 

 putations with accuracy, and places of decimals may be 

 freely cut off to save arithmetical work. The mean 

 equations having been computed, their solution by the 

 ordinary methods of algebra gives the most probable 

 values of the unknown quantities. 



Works upon the Theory of Probability. 



Regarding the Theory of Probability and the Law of 

 Error as most important subjects of study for any one who 

 desires to obtain a complete comprehension of scientific 

 method as actually applied in physical investigations, I 

 will briefly indicate the works in one or other of which 

 the reader will best pursue the study. 



The best popular, and at the same time profound English 

 work on the subject is De Morgan's " Essay on Proba- 

 bilities and on their Application to Life Contingencies and 

 Insurance Offices," published in the Cabinet Cyclopaedia, 

 and to be obtained (in print) from Messrs. Longman. 

 Mr. Venn's work on The Logic of Chance can now be 

 procured in a greatly enlarged second edition ; l it contains 

 a most interesting and able discussion of the metaphysical 



1 The Logic of Chance, au Essay on the Foundations and Province 

 of the Theory of Probability, with especial reference to its Logical 

 Bearings and its Application to Moral and Social Science. (Mac- 

 millan), 1876. 



