THE IRRIGATION AGE. 



823 



2 20' = cotg 87 40', etc., hence the trigonometrical tables 

 have been prepared so that every function for every angle 

 is the cofunction of the complement. 



3. Transmutation of Functions. 



In Fig. 36, when c 1, then a is the sine and b the 

 cosine of ; and since a" + b' = 1, then 



t>- and b = b V 



"' so tnat 



cos 2 a and 



cos r= Y/ l sin' a 



also tg <* = a/b = sin -^- cos a 



also cotg a = 6/a = cos -r- sin a 



also tg a = 1/cotg <* and 



COtg a = 1/tg a 



So to express the tangent a in sin a, substitute value ot 



cosine as above : __ 



tg a = sin a H- V 1 sin s a. or 

 tg a = \/ 1 Cos 2 " -T- Cos : also 

 cotg o = cos o -r- V 1 cos a :l 

 cotga=r -y/ 1 sin' a -H sin a. 

 4. The Signs of the Functions. 



In Fig. 37 a circle is divided into 4 

 equal sections, called quadrants, let the 

 radius OA equal 1 and let a moveable 

 radius OE be journalled at point O, so 

 that as it turns from the position OA 

 upward through the 4 quadrants it will 

 describe all the angles possible from 

 zero to 360 ; the angles in the first 

 quadrant run from to 90 ; in the 

 Fig. 37. second quadrant from 90 to 180, in 



the third quadrant from 180 to 270, and in the 4th quadrant 

 from 270 to 360; but the trigonometrical functions are 

 numerically the same in all the quadrants; thus if the angle 

 a 10 then its sine E F is also the sine of 170 , the sine 

 of 190 and the sine of 350, but the signs of the functions 

 are differing in the different quadrants; hence the functions 

 of the angles have been computed only for one quadrant. 



The signs of the various functions are indicated in the 

 following table : 



It i- -een that all the functions are positive in the first 

 quadrant and that in the second quadrant the sine is positive, 

 but the cosine tangent and cotangent are negative. 



The signs of the secant are the same as that of the cosine, 

 and the signs of the cosecant the same as that of the sine. 



5. Numerical Limits of. the Functions. 



By watching the moving radius in Fig. 37 it is seen 

 that E F = when a = o, so the sine of angle of 

 = zero and grows as the angle grows, and reaches a maximum 

 of 1 when a = 90 ; the cosine -of = 1 when a = , and 

 decreases as the angle increases, reaching when a = 

 Likewise the tg a = when a = ; the tangent grows 

 as the angle grows and becomes infinitely large when o 



The cotangent is the reverse of the tangent; when 

 the cotangent = o (infinitely large), as the angle 

 grows the cotangent decreases, and when a =- 90 the co- 



"^"is al0 seen that when a = 45, then the tangent and 

 cotangent are each equal to 1 and the sine and cosine each 

 equal to ^/ .5 = .7071. In a 60 triangle the cosine of an 

 an.cle of 60 = */ 2 . 



-IfMicaticn to Practical Problems. 



In the right angled triangle 

 ABC, Fig. 38, let AB = 17 ft., 

 AC = 12 ft.; find side EC and 

 aneles a and f$. 



Solution. The sine of the 

 angle B = AC/AB or 12/17 = 

 .70588; look in a table of 

 natural sines and opposite 

 .70587 you find 44 23', which 

 is the right angle to the near- 



Fig. 38. 



is the right angle to me near- 

 est minute : then since 0+3 = 90, then a. = 90 g, or 

 o = 90 40 23' = 45 P>7'. 



(Continued on page 833.) 



Supreme Court Decisions 



irrigation Cases 



DAMAGES FOR FLOODING LANDS 



The measure of damages for injury to land, by its 

 being covered with sediment and debris by being flooded 

 by the breaking of a reservoir, is the difference between 

 the value of the land before and immediately after the 

 injury and not the reasonable cost of clearing the land, 

 though evidence of such cost would be admissible to aid 

 in determining such difference in the value of the land. 

 Mustang Reservoir, Canal & Land Co. v. Hissman. Supreme 

 Court of Colorado. 113 Pacific 800. 

 DAMAGES FOR FLOODING LANDS 



In an action for damages by the flooding of plaintiff's 

 land by the breaking of defendant's reservoir, evidence 

 that about eight or ten months after the flooding plain- 

 tiff expended a certain sum in repairing his reservoir 

 embankment was not of itself sufficient to authorize the 

 allowance of damages on that ground; there being no evi- 

 dence to show that the repairs were made necessary by 

 the flooding, or that the amount expended was reason- 

 able, or as to the actual damage to the reservoir embank- 

 ment. Mustang Reservoir, Canal & Land Co. \. Hissman. 

 Supreme Court of Colorado. 112 Pacific 800. 

 RESERVATION OF WATER 



Plaintiff whose land was bordered by a stream con- 

 veyed water to his residence by means of a flume and 

 thereafter he conveyed a strip of land bordering on the 

 stream, and the deed reserved the right to "take and use 

 the water as now taken and used" and the right to re- 

 pair and maintain flumes "necessary to conduct the amount 

 of water now accustomed to flow across the lands con- 

 veyed." Held, that plaintiff's rights were limited to the 

 amount of water used by him at the time of conveyance, 



irrespective of the amount that flowed _in the flume. 



Peifrson v. Cody. Court of Appeal, First District, Cali- 

 fornia. 112 Pacific 558. 

 REASONABLE CHARGE 



What is a reasonable charge for irrigation within Laws 

 1907, chap. 49, regulating the use and distribution of water 

 for irrigation, and providing that the charges therefor shall 

 be reasonable, depends largely on the cost of constructing and 

 operating the irrigation works, and the price which the owners 

 of land can afford to pay for irrigation must depend in part 

 on the use to which the land can be put. Young & Norton v. 

 Hinderlider. Supreme Court of New Mexico. 110 Pacific 

 1045. 

 DOCTRINE OF APPROPRIATION IN COLORADO 



In Colorado, the common-law doctrine in respect of the 

 rights of riparian proprietors never has obtained, and in its 

 stead there was adopted the doctrine of appropriation which 

 regards the waters of all natural streams as subject to appro- 

 priation and diversion for beneficial uses and treats priority 

 of appropriation, and continued beneficial use as giving the 

 "jrior and better right. Snyder v. Colorado Gold Dredging Co. 

 U. S. Circuit Court of Appeals. 181 Federal 62. 

 IRRIGATION CORPORATION'S LIABILITY 



A corporation organized under Sayles' Ann Civ. St. 1897. 

 art. 3125, providing for irrigation corporations with power to 

 sell water to consumers pro rata, without preferences, which 

 contracts with a person entitled to water for irrigation to 

 furnish him with a full supply of water to make a crop, can- 

 not excuse nonperformance by proving shortage in the supply 

 from drought, accident, or other causes, so that it cannot com- 

 ply with the contract without discriminating against other 

 consumers of water. Erp v. Rayivood Canal & Milling Co. 

 Court of Civil Appeals of Texas. 130 Southwestern 897. 

 PATENT FOR GOLD PLACER AND UNAPPROPRIATED WATER 



In so far as the rights and incidents of riparian proprietor- 

 ship are concerned, conveyances by the United States of 

 public lands on nonnavigable streams and lakes, when it is 

 not provided otherwise, are to be construed and have effect 

 according to the law of the state in which the lands are 

 situate ; and by the law of Colorado a conveyance of riparian 

 land, even if it be a gold placer claim, does not carry any right 

 to the unappropriated waters of the stream. Snyder v. Colo- 



