The Compound Law of Error. 207 



loudest, being inaudible. This is surely a reductio ad absurdum 

 of the received theory. 



I do not wish to be understood as denying that the theory 

 has any basis of truth. My contention is that the actions to 

 which it is truly applicable play only a subordinate part in 

 the production of resultant tones. 



Appendix. 



Examples selected from Kcenig's Experiences d'Acoustique, 

 pp. 103 and 104, illustrating the production of the 

 common fundamental. The " single vibrations " of 

 the original are here reduced to double or complete 

 vibrations : — 

 ut 5 and s%, which are as 8 : 15, gave only ut%. 

 ut 5 and 2816, which are as 4 : 11, gave ut B corresponding 

 to 1 louder than any other tone. 



ut 5 and si 6 , which are as 4 : 15, gave no audible tone but ut 3 . 

 ut 5 and 3968, which are as 8 : 31, gave no audible tone but 

 ut 2 . 



ut 6 and 3584, which are as 4 : 7, gave ut± more distinct than 

 the difference-tone sol 5 . 



ut 6 and si 6 , which are as 8 : 15, gave ut B distinct, the diffe- 

 rence-tone 7 being inaudible. 



ut 6 and 3968, which are 16 : 31, gave ut 2 only. 

 ut 6 and 4032, which are 32 : 63, gave ut^ only. 



XXY. The Compound Law of Error. 

 By Professor F. Y. Edgeworth, M.A., D.C.L.* 



THE compound law of error is an extension to the case of 

 several dimensions of the simple law for the frequency 

 with which a quantity of one dimension (x) tends to assume 

 each particular value. A first approximation to the com- 

 pound law has been obtained by several writers independently, 

 —by Mr. De Forest, in the 'Analyst' for 1881; by the 

 present writer, in the Philosophical Magazine for December 

 1892 ; and by Mr. S. IL Burbury, in the same Journal for 

 January 1894. I propose here to employ tbe method of 

 partial differential equations explained in a preceding paper | 

 to verify the first approximation, and to discover a second 

 approximation, to the compound law. 



To begin with the case of two dimensions : let Q be the 



* Communicated by the Author. 



t " On the Asymmetrical Probability-Curve," Phil, Mag. February 1896. 



