INSTRUMENTS AND METHODS OF RESEARCH 191 



The irregularity of the banks bordering a natural watercourse serves 

 to differentiate the work of nature from that of the builder of the 

 artificial and regular channel. 



No, instead of rejecting, we must learn to retain the outstanding 

 residuals and study them most carefully and regard them as the true 

 facts of nature, and not those which we so egotistically and presump- 

 tuously try to force on her. What great discoveries may lie open 

 to us when we once have grasped the true significance of the facts 

 we have been so fond of measuring by our own standard and have 

 been terming as " abnormal " or " irregular " ! 



An interesting example of not wholly successful application of the 

 continuous and ever-recurring functions of spherical harmonics to a 

 typical geophysical phenomenon — the distribution of magnetism over 

 the earth's surface — has been discussed by the speaker elsewhere. 

 Though the number of unknowns has been increased in recent com- 

 putations from the original 24 of Gauss to 48, nevertheless the dif- 

 ference between theory and observation is of such an order of magni- 

 tude as to preclude the use of the formula for even the purely 

 practical demands of the navigator and surveyor. Nor has any one 

 succeeded in giving any physical interpretation of the laboriously de- 

 rived coefficients beyond the first three. And what do these three 

 stand for? The simplest possible case of a first approximation to 

 the actual state of the earth's magnetism, viz., that of a uniform 

 magnetization about a diameter inclined to the axis of rotation ! 



The prime difficulty here may be summed up in a word. The very 

 surface over which the spherical harmonic functions are spread is it- 

 self such a prolific source of disturbance as to cause effects embracing 

 a continent, a state or a locality. Such a large number of terms 

 would be requisite for an adequate representation as to make their 

 computation prohibitive. We are dealing here with more or less non- 

 continuous effects that cannot be imitated by continuous functions 

 without leaving behind a train of residuals, precisely as though we 

 were to try to fit to the actual configuration of the earth some stand- 

 ard pattern of our own. Let me ask what phenomenon have we, in 

 fact, which will admit of the determination of 48, or even of 24, 

 physical constants? 



It had been my intention to say a few words on the value and 

 limitation of that much-used as well as abused mathematical instru- 

 ment of research, the method of least squares. Properly employed, 

 it is a most useful adjunct to investigation; but, as intimated, the 

 true significance of formulae established by this method is at times 

 pushed far beyond the limitations. What the tenor of my remarks 

 might be will be sufficiently evident to you if I submit this query for 



