This force has been reduced by detachments to Johnston. It is as well known as we can ever expect to ascertain such facts, that three brigades have gone from McConn's division, and two or three from Breckinridge's,--say two. It is clear that there are now but four infantry divisions in Bragg's army, the fourth being composed of fragments of McConn's and Breckinridge's divisions, and must be much smaller than the average. Deducting the five brigades, and supposing them composed of only four regiments each, which is below the general average, it gives an infantry reduction of twenty regiments, four hundred each--eight thousand--leaving a remainder of thirty thousand. It is clearly ascertained that at least two brigades of cavalry have been sent from Van Dorn's command to the Mississippi, and it is asserted in the Chattanooga _Rebel_, of June 11th, that General Morgan's command has been permanently detached and sent to eastern Kentucky. It is not certainly known how large his division is, but it is known to contain at least two brigades. Taking this minimum as the fact, we have a cavalry reduction of four brigades. Taking the lowest estimate, four regiments to the brigade, we have a reduction, by detachment, of sixteen regiments, five hundred each, leaving his present effective cavalry force nine thousand five hundred. With the nine brigades of the two arms thus detached, it will be safe to say there have gone,-- Six batteries, 80 men each, 480 Leaving him 20 batteries, 2,120 Making a total reduction of 16,480 Leaving, of the three arms, 41,680 In this estimate of Bragg's strength, I have placed all doubts in his favor, and I have no question that my estimate is considerably beyond the truth. General Sheridan, who has taken great pains to collect evidence on this point, places it considerably below these figures. But assuming these to be correct, and granting what is still more improbable, that Bragg would abandon all his rear posts, and entirely neglect his communications, and could bring his last man into battle, I next ask: What have we with which to oppose him? The last official report of effective strength now on file in the office of the assistant adjutant-general, is dated from June 11th, and shows that we have in this department, omitting all officers and enlisted men attached to department, corps, division and brigade headquarters,-- 1. Infantry--One hundred and seventy-three regiments; ten battalions sharp-shooters; four battalions pioneers; and one regiment of engineers and mechanics, with a total effective strength of seventy thousand nine hundred and eighteen. 2. Cavalry--Twenty-seven regiments and one unattached company, eleven thousand eight hundred and thirteen. 3. Artillery--Forty-seven and a half batteries field artillery, consisting of two hundred and ninety-two guns and five hundred and sixty-nine men,--making a general total of eighty-seven thousand eight hundred.