INSTRUMENTS AND METHODS OF RESEARCH. 
113 
magnitude 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 interpre- 
tation of the laboriously derived 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 func- 
tions are spread is itself 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 pro- 
hibitive. We are dealing here with more or less discontinu- 
ous 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 standard 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 mathe- 
matical instrument of research, the method of least squares. 
Properly employed, it. is a most useful adjunct to investi- 
gation ; but, as intimated, the true significance of formulae 
established by this method is at times pushed way beyond 
the limitations. What the tenor of my remarks might be 
will be sufficiently evident to you if I submit this query for 
your consideration: What actual laics of Nature have been 
discovered by the method of least squares? 
The Mechanical Instruments of Research. 
A few minutes were to have been given to the instruments 
employed by the scientific man to sharpen and amplify his 
natural senses and sensations — in a word, the tools furnished 
him by the mechanician. I am glad, however, both for 
