MATHEMATICAL THEORY OE STREAM-LINES. 
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Augmented surface, f-|vr 2 £ 2 = 25/ 2 nearly ; 
Girth-integral, §Gdx = T+ 2 = 9 • 8 7 Z 2 nearly ; 
Coefficient of augmentation, ^-^=2*531 nearly. 
In each case 0 denotes the angle made by a given radius with the direction of motion. 
For a sphere half-immersed the augmented surface and girth-integral have respect- 
ively half the values given above. 
For an approximately trochoidal riband of uniform breadth, it has been elsewhere 
shown (Philosophical Transactions, 1863, p. 134) that the coefficient of augmentation is 
very nearly 1+4 sin 2 /3+ sin 4 /3, (3 being the angle of greatest obliquity of the riband to 
the direction of motion. 
With a view to the calculation of the augmented surface by numerical definite inte- 
gration in particular cases, the following values of the elementary surface da and of its 
first integral are given. As to the function see § 3. 
General case: 
da— V { dy 2 dz t -j- dz 2 dx 2 + dx 2 dy l } (88) 
Cylindrical surface of indefinite depth ; da per unit of depth = .... (88 a) 
Surface of revolution, half -immersed ; fla for a zone or belt measuring dx 1 
1 fl • iryqdx , (8>S B) 
lengthwise = -Ar — I 
§ 20. General Remarks . — The dynamical investigations contained in this chapter are 
partly certain and exact, partly approximate, and partly conjectural. The results 
arrived at in §§ 12 to 15 as to momentum and energy of current and of disturbance, are 
all certain and exact when applied to the case of a solid body of any figure past which 
a fluid can glide continuously, immersed in an unlimited mass of liquid, and ap- 
proximate when applied to cases such as those described in § 16, in which these con- 
ditions are approximately fulfilled. The results as to virtual depth of disturbance, and 
as to speed of waves, in § 17, are partly exact, and partly approximate. The probable 
laws of wave-resistance and of skin-resistance, in §§ 18 and 19, are partly conjectural, 
and require the aid of much additional experimental research to test and verify them, 
and to make them definite ; but still they have already to a certain extent been verified 
by observations of the performance of ships. The whole body of results, whether certain 
or conjectural, are set forth in the hope that they may prove useful in deducing general 
principles from the data of experiment and observation, and in suggesting plans for 
further research. 
