Tidal Phenomena at Inland Boreholes near Gradock. 
87 
evidence that the Tarka Bridge tide is produced as a consequential effect 
of oceanic tides. The connection between the two may be merely that of 
a common astronomical cause. 
Determination of the various Harmonic Periods in the Tarka Bridge 
Curve. 
The advisability of subjecting the Tarka Bridge records to a harmonic 
analysis occurred to my mind at an early stage of my investigations. 
On inquiring into the methods of harmonic analysis commonly 
adopted in the study of marine tide curves I found that these methods are 
based on assumptions that the curves consist of harmonic components the 
periods of which are known or assumed from astronomically observed 
data. As my object was to prove or disprove a definite connection between 
the harmonic components of the curve and astronomical data, I concluded 
that the above-mentioned methods were inapplicable for my purpose. 
Lately there came under my notice a method invented by Prof. Chrystal 
for the determination of the periods of the harmonic components of limno- 
grams or seiche curves. This is a method which is based on no assump- 
tions as to the nature of the causes producing the movements indicated on 
the curves, and a careful consideration of the mathematical theory of the 
method convinced me that it could be applied with perfect confidence to 
the Tarka Bridge records. Chrystal speaks of it as the "Method of Eesi- 
duation," and its theory and mode of application is described fully in 
Trans. Eoy. Soc, Edinburgh, vol. xlv., part 2, pp. 385-7. As this 
reference is not easily accessible to most South African readers, and as the 
method is little known, I venture to quote it in full in an Appendix. 
From the form in which Chrystal gives his mathematical theory it may 
not be immediately obvious that the method is applicable to the Tarka 
Bridge curves, and the following explanation may perhaps be necessary. 
Fourier has shown that any finite periodic function of a variable can be 
expressed as a series of terms, each of which is a simple harmonic function 
of this variable. 
Thus if y be any periodic function of the time, y can by suitably 
choosing the constants A, M^, M2, &c., <3j, e^, &c., be expressed by a series 
of terms as follows — 
2/ = A + Mi sin {kt + e,) + M^ sin {2kt + e^) + &c. 
Without making any further assumptions as to the nature of the 
function this equation can be transformed into the form — 
Stt . 27r 
?/ = Aj sin TTT {t — a^) + k^ sin ^ (t — a^) + kQ., 
