Fundamental Principles of Scientific Inquiry. 385 



For instance, the class of all functions whatever, and the 

 class of all continuous functions, have each the number C c ; 

 even the class of analytic functions has the number C 

 or 2*o, which is greater than K . We cannot therefore 

 admit the possibility of every analytic function. Even if 

 the coefficients in the expansions of analytic functions in 

 power series are restricted to be rational numbers, the 

 number of possibilities is still 2 K °. We are therefore 

 reduced to trying to specify a class of functions, tt in 

 number, which will include all functions at present known 

 to physics. The class of all polynomials with rational or 

 algebraic coefficients has the proper number ; but it does 

 not fulfil the second condition, since it does not include 

 trigonometric functions. The same applies to the class 

 of algebraic functions involving no constants other than 

 rational and algebraic numbers. Non-algebraic coefficients 

 are, of course, not all admissible, since the number of 

 possibilities would at once rise to 2 H °. Some transcendental 

 functions must be admitted if our assumptions are to cover 

 our present beliefs, which we are not prepared to abandon 

 without very strong reason. We may notice, however, 

 that these transcendental functions are hardly ever derived 

 directly from observation ; they practically always arise 

 first in theoretical work, and it is not till afterwards that 

 it is verified that they satisfy the results obtained by obser- 

 vation. In the theoretical work they arise as the solutions 

 of differential equations of finite order and degree. The 

 coefficients in these, however, usually involve constants, 

 such as mass, electric charge, and size, which are appa- 

 rently more or less subject to our own control ; if these 

 are supposed capable of continuous variation, a differential 

 equation involving any of them is only one of a clnss whose 

 number is 0, and our suggestion fails. This difficulty is 

 readily overcome in either of two ways. We can suppose 

 that these quantities can only vary by finite stage?, in 

 which case the number of possible values is only K ; or 

 we can suppose that any equation involving them is not 

 in its most fundamental form, and that it should be dif- 

 ferentiated again and have these constants eliminated. 

 The former suggestion agrees with modern ideas on the 

 discontinuous structure of matter ; the latter is tenable 

 even if these ideas should be abandoned. Now the number 

 of differential equations of finite order and degree with 



Phil. Mag. S. 6. Vol. 42. No. 249. Sept. 1021. 2 



